# L-Functions

- [Overview](overview.md)

- [Built-in $L$-series](builtin.md)

  - [`RiemannZeta() → LSer`](builtin.md#function-riemannzeta)

  - [`Example: Lseries Sig Riemann`](builtin.md#example-ex-b6ea59)

  - [`LSeries(K): FldNum → LSer`](builtin.md#function-lseries-fldnum)

  - [`Example: Lseries Sig Dedekind`](builtin.md#example-ex-a7a8a2)

  - [`Example: Lseries Sig Dedekind2`](builtin.md#example-ex-e875eb)

  - [`Example: armitage`](builtin.md#example-ex-f2ca95)

  - [`LSeries(A): ArtRep → LSer`](builtin.md#function-lseries-artrep)

  - [`Example: Lseries Artin`](builtin.md#example-ex-e5651d)

  - [`Example: Lseries A7`](builtin.md#example-ex-c074bd)

  - [`LSeries(E): CrvEll → LSer`](builtin.md#function-lseries-crvell)

  - [`Example: Lseries Sig Elliptic`](builtin.md#example-ex-f74234)

  - [`LSeries(E, K): CrvEll, FldNum → LSer`](builtin.md#function-lseries-crvell-fldnum)

  - [`Example: Lseries Sig Ellnf`](builtin.md#example-ex-9cb4a0)

  - [`LSeries(E, A): CrvEll, ArtRep → LSer`](builtin.md#function-lseries-crvell-artrep)

  - [`Example: Lseries Sig Ellartintwist`](builtin.md#example-ex-50ee8e)

  - [`Example: Lseries Etw Quaternion`](builtin.md#example-ex-e42206)

  - [`LSeries(C): CrvHyp[FldRat] → LSer`](builtin.md#function-lseries-crvhyp-fldrat)

  - [`Example: Lseries Sig Crvhyp`](builtin.md#example-ex-8fc046)

  - [`LSeries(C,K): CrvHyp[FldRat], FldNum → LSer`](builtin.md#function-lseries-crvhyp-fldrat-fldnum)

  - [`Example: Lseries Crvhyp Qnf`](builtin.md#example-ex-1a6295)

  - [`LSeries(C): CrvHyp[FldNum] → LSer`](builtin.md#function-lseries-crvhyp-fldnum)

  - [`Example: Lseries Crvhyp Nf`](builtin.md#example-ex-73aef0)

  - [`LSeries(Chi): GrpDrchElt → LSer`](builtin.md#function-lseries-grpdrchelt)

  - [`Example: Lseries Sig Character`](builtin.md#example-ex-ac2d65)

  - [`LSeries(hmf): ModFrmHilElt → LSer`](builtin.md#function-lseries-modfrmhilelt)

  - [`Example: Lseries Hilbert Modform`](builtin.md#example-ex-846e3d)

  - [`LSeries(omf): ModFrmAlgElt → LSer`](builtin.md#function-lseries-modfrmalgelt)

  - [`Example: Lseries Orthogonal Modform`](builtin.md#example-ex-cc704a)

  - [`LSeries(psi): GrpHeckeElt → LSer`](builtin.md#function-lseries-grpheckeelt)

  - [`LSeries(psi): GrossenChar → LSer`](builtin.md#function-lseries-grossenchar)

  - [`LSeries(f): ModFrmElt → LSer`](builtin.md#function-lseries-modfrmelt)

  - [`Example: Lseries Sig Modfrm`](builtin.md#example-ex-0f42a0)

  - [`LSeries(S): ModSym → LSer`](builtin.md#function-lseries-modsym)

  - [`Example: Lseries Sig Modsym`](builtin.md#example-ex-52ab37)

- [Computing $L$-values](computing.md)

  - [`Evaluate(L, s0): LSer, FldComElt → FldComElt`](computing.md#function-evaluate-lser-fldcomelt)

  - [`CentralValue(L): LSer → FldComElt`](computing.md#function-centralvalue-lser)

  - [`LStar(L, s0): LSer, FldComElt → FldComElt`](computing.md#function-lstar-lser-fldcomelt)

  - [`LTaylor(L,s0,n): LSer, FldComElt, RngIntElt → FldComElt`](computing.md#function-ltaylor-lser-fldcomelt-rngintelt)

  - [`Example: Lseries Evaluate`](computing.md#example-ex-7d80f5)

  - [`DedekindZetaExact(K, z): Fld, RngIntElt → FldRatElt`](computing.md#function-dedekindzetaexact-fld-rngintelt)

- [General $L$-series](general.md)

  - [Terminology](general.md#terminology)

  - [Constructing a General $L$-Series](general.md#constructing-a-general-l-series)

    - [`LSeries({weight, gamma, conductor, }{cffun}): FldReElt, [FldRatElt], FldReElt, Any → LSer`](general.md#function-lseries-fldreelt-fldratelt-fldreelt-any)

    - [`LSeries(HS, N, cffun): HodgeStruc, RngIntElt, Any → LSer`](general.md#function-lseries-hodgestruc-rngintelt-any)

    - [`CheckFunctionalEquation(L): LSer → FldComElt`](general.md#function-checkfunctionalequation-lser)

    - [`CFENew(L): LSer → FldReElt`](general.md#function-cfenew-lser)

    - [`Example: Lseries Checkfun`](general.md#example-ex-37a053)

  - [Setting the Coefficients](general.md#setting-the-coefficients)

    - [`LSetCoefficients(L,cffun): LSer, Any → {}`](general.md#function-lsetcoefficients-lser-any)

  - [Specifying the Coefficients Later](general.md#specifying-the-coefficients-later)

    - [`Example: Lseries Lcfrequired`](general.md#example-ex-147e8a)

  - [Generating the Coefficients from Local Factors](general.md#generating-the-coefficients-from-local-factors)

- [Accessing the Invariants](invariants.md)

  - [`LCfRequired(L): LSer → RngIntElt`](invariants.md#function-lcfrequired-lser)

  - [`LGetCoefficients(L, N): LSer, RngIntElt → List`](invariants.md#function-lgetcoefficients-lser-rngintelt)

  - [`EulerFactor(L, p): LSer, RngIntElt → RngElt`](invariants.md#function-eulerfactor-lser-rngintelt)

  - [`Degree(L): LSer → RngIntElt`](invariants.md#function-degree-lser)

  - [`Conductor(L): LSer → RngElt`](invariants.md#function-conductor-lser)

  - [`Sign(L): LSer → RngElt`](invariants.md#function-sign-lser)

  - [`MotivicWeight(L): LSer → RngIntElt`](invariants.md#function-motivicweight-lser)

  - [`GammaFactors(L): LSer → SeqEnum`](invariants.md#function-gammafactors-lser)

  - [`LSeriesData(L): LSer → Info`](invariants.md#function-lseriesdata-lser)

  - [`BadPrimeData(L): LSer → SeqEnum`](invariants.md#function-badprimedata-lser)

  - [`Example: Lseries Invariants 1`](invariants.md#example-ex-f37c5f)

  - [`Factorization(L): LSer → SeqEnum[Tup]`](invariants.md#function-factorization-lser)

  - [`Factorisation(L): LSer → SeqEnum[Tup]`](invariants.md#function-factorisation-lser)

  - [`Example: Lseries Invariants 2`](invariants.md#example-ex-52681e)

- [Modifying the $L$-function](modifyL.md)

  - [`ChangeEulerFactor(L,p,f): LSer, RngIntElt, RngUPolElt → LSer`](modifyL.md#function-changeeulerfactor-lser-rngintelt-rngupolelt)

  - [`ChangeLocalInformation(L,p,d,f): LSer, RngIntElt, RngIntElt, RngUPolElt → LSer`](modifyL.md#function-changelocalinformation-lser-rngintelt-rngintelt-rngupolelt)

  - [`ChangeLocalInformation(L,bp): LSer, List → LSer`](modifyL.md#function-changelocalinformation-lser-list)

  - [`CopyCoefficients(L,M): LSer, LSer`](modifyL.md#function-copycoefficients-lser-lser)

  - [`Example: Change Local Info`](modifyL.md#example-ex-05dc99)

- [Precision](precision.md)

  - [`LSetPrecision(L,precision): LSer, RngIntElt → {}`](precision.md#function-lsetprecision-lser-rngintelt)

  - [$L$-series with Unusual Coefficient Growth](precision.md#l-series-with-unusual-coefficient-growth)

  - [Computing $L(s)$ when Im$(s)$ is Large (ImS Parameter)](precision.md#computing-l-s-when-im-s-is-large-ims-parameter)

  - [Implementation of $L$-series Computations (Asymptotics Parameter)](precision.md#implementation-of-l-series-computations-asymptotics-parameter)

- [Verbose Printing](verbose.md)

- [Arithmetic with $L$-series](arithmetic.md)

  - [`L1 * L2: LSer, LSer → LSer`](arithmetic.md#operation-op-times-lser-lser)

  - [`L1 / L2: LSer, LSer → LSer`](arithmetic.md#operation-op-div-lser-lser)

  - [Hodge Structure](arithmetic.md#hodge-structure)

    - [`HasHodgeStructure(L): LSer → BoolElt, HodgeStruc`](arithmetic.md#function-hashodgestructure-lser)

    - [`HodgeStructure(L): LSer → HodgeStruc`](arithmetic.md#function-hodgestructure-lser)

    - [`HodgeStructure(X): SeqEnum → HodgeStruc`](arithmetic.md#function-hodgestructure-seqenum)

    - [`HodgeStructure(w, G): RngIntElt, SeqEnum → HodgeStruc`](arithmetic.md#function-hodgestructure-rngintelt-seqenum)

    - [`Dual(HS): HodgeStruc → HodgeStruc`](arithmetic.md#function-dual-hodgestruc)

    - [`TateTwist(HS, k): HodgeStruc, RngIntElt → HodgeStruc`](arithmetic.md#function-tatetwist-hodgestruc-rngintelt)

    - [`Translate(L, z): LSer, RngIntElt → LSer`](arithmetic.md#function-translate-lser-rngintelt)

    - [`Translate(L, z): LSer, FldRatElt → LSer`](arithmetic.md#function-translate-lser-fldratelt)

    - [`GammaFactors(HS): HodgeStruc → SeqEnum`](arithmetic.md#function-gammafactors-hodgestruc)

    - [`GammaShifts(HS): HodgeStruc → SeqEnum`](arithmetic.md#function-gammashifts-hodgestruc)

    - [`Degree(HS): HodgeStruc → RngIntElt`](arithmetic.md#function-degree-hodgestruc)

    - [`Weight(HS): HodgeStruc → RngIntElt`](arithmetic.md#function-weight-hodgestruc)

    - [`EffectiveHodgeStructure(HS): HodgeStruc → HodgeStruc, RngIntElt`](arithmetic.md#function-effectivehodgestructure-hodgestruc)

    - [`RootNumber(HS): HodgeStruc → FldCycElt`](arithmetic.md#function-rootnumber-hodgestruc)

    - [`TensorProduct(H1, H2): HodgeStruc, HodgeStruc → HodgeStruc`](arithmetic.md#function-tensorproduct-hodgestruc-hodgestruc)

    - [`SymmetricPower(HS, m): HodgeStruc, RngIntElt → HodgeStruc`](arithmetic.md#function-symmetricpower-hodgestruc-rngintelt)

    - [`Determinant(HS): HodgeStruc → HodgeStruc`](arithmetic.md#function-determinant-hodgestruc)

    - [`AlternatingSquare(HS): HodgeStruc → HodgeStruc`](arithmetic.md#function-alternatingsquare-hodgestruc)

    - [`HodgeVector(HS): HodgeStruc → SeqEnum, RngIntElt`](arithmetic.md#function-hodgevector-hodgestruc)

    - [`CriticalPoints(HS): HodgeStruc → SeqEnum`](arithmetic.md#function-criticalpoints-hodgestruc)

    - [`CriticalPoints(L): LSer → SeqEnum`](arithmetic.md#function-criticalpoints-lser)

    - [`ImaginaryTwist(HS): HodgeStruc → HodgeStruc`](arithmetic.md#function-imaginarytwist-hodgestruc)

    - [`Example: Lseries Hodge Struc`](arithmetic.md#example-ex-271a08)

  - [Tensor Products](arithmetic.md#tensor-products)

    - [`TensorProduct(L1, L2, ExcFactors): LSer, LSer, [<>] → LSer`](arithmetic.md#function-tensorproduct-lser-lser)

    - [`TensorProduct(L1, L2, ExcFactors, K): LSer, LSer, [<>], FldNum → LSer`](arithmetic.md#function-tensorproduct-lser-lser-fldnum)

    - [`TensorProduct(L1, L2): LSer, LSer → LSer`](arithmetic.md#function-tensorproduct-lser-lser-2)

    - [`TensorProduct(L1, L2, K): LSer, LSer, FldNum → LSer`](arithmetic.md#function-tensorproduct-lser-lser-fldnum-2)

    - [`Example: Ec Tensorprod`](arithmetic.md#example-ex-8a6681)

    - [`Example: Level1 Modform`](arithmetic.md#example-ex-0e486d)

    - [`Example: Siegel Modular Form`](arithmetic.md#example-ex-f0736e)

    - [`Example: Tensprod OverK`](arithmetic.md#example-ex-f991ed)

    - [`Example: Consani Scholten`](arithmetic.md#example-ex-1ad661)

  - [Symmetric Powers](arithmetic.md#symmetric-powers)

    - [`Determinant(L): LSer → LSer`](arithmetic.md#function-determinant-lser)

    - [`SymmetricPower(L, m): LSer, RngIntElt → LSer`](arithmetic.md#function-symmetricpower-lser-rngintelt)

    - [`Example: Lseries Sympow`](arithmetic.md#example-ex-2b6618)

    - [`Example: Sympow Gross`](arithmetic.md#example-ex-833dcd)

    - [`Example: Sympow Ec`](arithmetic.md#example-ex-b9aa14)

    - [`Example: Sympow Ec2`](arithmetic.md#example-ex-5c1fea)

    - [`Symmetrization(L, p): LSer, SeqEnum → LSer`](arithmetic.md#function-symmetrization-lser-seqenum)

    - [`IsOrthogonal(L): LSer → BoolElt`](arithmetic.md#function-isorthogonal-lser)

    - [`IsSymplectic(L): LSer → BoolElt`](arithmetic.md#function-issymplectic-lser)

    - [`OrthogonalSymmetrization(L, p): LSer, SeqEnum → LSer`](arithmetic.md#function-orthogonalsymmetrization-lser-seqenum)

    - [`SymplecticSymmetrization(L, p): LSer, SeqEnum → LSer`](arithmetic.md#function-symplecticsymmetrization-lser-seqenum)

    - [`Example: General Symm`](arithmetic.md#example-ex-631189)

    - [`Example: Orthog Symm`](arithmetic.md#example-ex-401e29)

    - [`Example: Symplectic Symm`](arithmetic.md#example-ex-d40acc)

    - [`Example: More Orthog`](arithmetic.md#example-ex-5e5bc4)

    - [`Example: Final Symm Examples`](arithmetic.md#example-ex-8645f8)

- [Advanced Examples](examples.md)

  - [Handmade $L$-series of an Elliptic Curve](examples.md#handmade-l-series-of-an-elliptic-curve)

    - [`Example: Lseries Elliptic Selfmade`](examples.md#example-ex-84defa)

  - [Self-made Dedekind Zeta Function](examples.md#self-made-dedekind-zeta-function)

    - [`Example: Lseries Dedekind Selfmade`](examples.md#example-ex-aa77ea)

  - [Handmade $L$-series of a Hyperelliptic Curve](examples.md#handmade-l-series-of-a-hyperelliptic-curve)

    - [`Example: Lseries Genus2`](examples.md#example-ex-d8bc35)

  - [Experimental Mathematics for Small Conductor](examples.md#experimental-mathematics-for-small-conductor)

    - [`Example: Lseries Experimental`](examples.md#example-ex-c04c9d)

  - [Tensor Product of $L$-series Coming from $l$-adic Representations](examples.md#tensor-product-of-l-series-coming-from-l-adic-representations)

    - [`Example: Lseries Tensor`](examples.md#example-ex-3815af)

  - [Non-abelian Twist of an Elliptic Curve](examples.md#non-abelian-twist-of-an-elliptic-curve)

    - [`Example: Lseries Nonabtwist`](examples.md#example-ex-3cebc3)

  - [Unitary Twist with Central Vanishing](examples.md#unitary-twist-with-central-vanishing)

    - [`Example: Rohrlich Example`](examples.md#example-ex-a7958b)