L-Functions
- Overview
- Built-in \(L\)-series
- Computing \(L\)-values
Evaluate(L, s0): LSer, FldComElt → FldComEltCentralValue(L): LSer → FldComEltLStar(L, s0): LSer, FldComElt → FldComEltLTaylor(L,s0,n): LSer, FldComElt, RngIntElt → FldComEltExample: Lseries EvaluateDedekindZetaExact(K, z): Fld, RngIntElt → FldRatElt
- General \(L\)-series
- Terminology
- Constructing a General \(L\)-Series
LSeries({weight, gamma, conductor, }{cffun}): FldReElt, [FldRatElt], FldReElt, Any → LSerLSeries(HS, N, cffun): HodgeStruc, RngIntElt, Any → LSerCheckFunctionalEquation(L): LSer → FldComEltCFENew(L): LSer → FldReEltExample: Lseries Checkfun
- Setting the Coefficients
- Specifying the Coefficients Later
- Generating the Coefficients from Local Factors
- Accessing the Invariants
- Modifying the \(L\)-function
ChangeEulerFactor(L,p,f): LSer, RngIntElt, RngUPolElt → LSerChangeLocalInformation(L,p,d,f): LSer, RngIntElt, RngIntElt, RngUPolElt → LSerChangeLocalInformation(L,bp): LSer, List → LSerCopyCoefficients(L,M): LSer, LSerExample: Change Local Info
- Precision
- Verbose Printing
- Arithmetic with \(L\)-series
L1 * L2: LSer, LSer → LSerL1 / L2: LSer, LSer → LSer- Hodge Structure
HasHodgeStructure(L): LSer → BoolElt, HodgeStrucHodgeStructure(L): LSer → HodgeStrucHodgeStructure(X): SeqEnum → HodgeStrucHodgeStructure(w, G): RngIntElt, SeqEnum → HodgeStrucDual(HS): HodgeStruc → HodgeStrucTateTwist(HS, k): HodgeStruc, RngIntElt → HodgeStrucTranslate(L, z): LSer, RngIntElt → LSerTranslate(L, z): LSer, FldRatElt → LSerGammaFactors(HS): HodgeStruc → SeqEnumGammaShifts(HS): HodgeStruc → SeqEnumDegree(HS): HodgeStruc → RngIntEltWeight(HS): HodgeStruc → RngIntEltEffectiveHodgeStructure(HS): HodgeStruc → HodgeStruc, RngIntEltRootNumber(HS): HodgeStruc → FldCycEltTensorProduct(H1, H2): HodgeStruc, HodgeStruc → HodgeStrucSymmetricPower(HS, m): HodgeStruc, RngIntElt → HodgeStrucDeterminant(HS): HodgeStruc → HodgeStrucAlternatingSquare(HS): HodgeStruc → HodgeStrucHodgeVector(HS): HodgeStruc → SeqEnum, RngIntEltCriticalPoints(HS): HodgeStruc → SeqEnumCriticalPoints(L): LSer → SeqEnumImaginaryTwist(HS): HodgeStruc → HodgeStrucExample: Lseries Hodge Struc
- Tensor Products
TensorProduct(L1, L2, ExcFactors): LSer, LSer, [<>] → LSerTensorProduct(L1, L2, ExcFactors, K): LSer, LSer, [<>], FldNum → LSerTensorProduct(L1, L2): LSer, LSer → LSerTensorProduct(L1, L2, K): LSer, LSer, FldNum → LSerExample: Ec TensorprodExample: Level1 ModformExample: Siegel Modular FormExample: Tensprod OverKExample: Consani Scholten
- Symmetric Powers
- Advanced Examples
