Computing \(L\)-values

Once an \(L\)-series \(L(s)\) has been constructed using either a standard \(L\)-function (Section Built-in \(L\)-series), a user defined \(L\)-function (Section Constructing a General \(L\)-Series) or constructed from other \(L\)-functions (Section Arithmetic with \(L\)-series), Magma can compute values \(L(s_0)\) for complex \(s_0\), values for the derivatives \(L^{(k)}(s_0)\) and Taylor expansions.

Evaluate(L, s0): LSer, FldComElt -> FldComElt

Derivative: RngIntElt                    Default: 0
Leading   : BoolElt                      Default: false

Given the \(L\)-series \(L\) and a complex number \(s_0\), the intrinsic computes either \(L(s_0)\), or if \(D>0\), the value of the derivative \(L^{(D)}(s_0)\). If \(D>0\) and it is known that all the lower derivatives vanish,

\[L(s_0)=L'(s_0)=...=L^{(D-1)}(s_0)=0\,,\]

the computation time can be substantially reduced by setting Leading:=true. This is useful if it is desired to determine experimentally the order of vanishing of \(L(s)\) at \(s_0\) by successively computing the first few derivatives.

CentralValue(L): LSer -> FldComElt

Given an \(L\)-function of motivic weight \(2k-1\), the value of \(L\) is computed at \(s=k\).

LStar(L, s0): LSer, FldComElt -> FldComElt

Derivative: RngIntElt                    Default: 0

Given the \(L\)-series \(L\) and a complex number \(s_0\), the intrinsic computes either the value \(L^*(s_0)\) or, if \(D>0\), the value of the derivative \(L^{*(D)}(s_0)\). Here \(L^*(s)=\gamma(s)L(s)\) is the modified \(L\)-function that satisfies the functional equation (cf. Section Terminology)

\[L^*(s) = {\rm sign} \cdot \bar L^*({\rm weight}-s)\]

(cf. Section Terminology).

LTaylor(L, s0, n): LSer, FldComElt, RngIntElt -> FldComElt

ZeroBelow: RngIntElt                    Default: 0

Compute the first \(n+1\) terms of the Taylor expansion of the \(L\)-function about the point \(s=s_0\), where \(s_0\) is a complex number:

\[L(s_0)+L'(s_0)x+L''(s_0)x^2/2!+\ldots+L^{(n)}(s_0)x^n/n!+O(x^{n+1}) \,\,.\]

If the first few terms \(L(s_0),...,L^{(k)}(s_0)\) of this expansion are known to be zero, the computation time can be reduced by setting ZeroBelow:=k+1.

Example: Lseries Evaluate (ex-7d80f5)

We define an elliptic curve \(E\) of conductor 5077 and compute derivatives at \(s=1\) until a non-zero value is reached:

> E := EllipticCurve([0, 0, 1, -7, 6]);
> L := LSeries(E : Precision:=15);
> Evaluate(L, 1);
0.000000000000000
> Evaluate(L, 1 : Derivative:=1, Leading:=true);
1.87710082755801E-24
> Evaluate(L, 1 : Derivative:=2, Leading:=true);
-6.94957228421048E-24
> Evaluate(L, 1 : Derivative:=3, Leading:=true);
10.3910994007158

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This suggests that \(L(E,s)\) has a zero of order 3 at \(s=1\). In fact, \(E\) is the elliptic curve (over Q) of smallest conductor with Mordell–Weil rank 3:

> Rank(E);
3

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Consequently, a zero of order 3 is predicted by the Birch–Swinnerton-Dyer conjecture. We can also compute a few terms of the Taylor expansion about \(s=1\), with or without specifying that the first three terms vanish.

> time LTaylor(L, 1, 5 : ZeroBelow:=3);
1.73184990011930*$.1^3 - 3.20590558844390*$.1^4 + 2.80009237167013*$.1^5 +
   O($.1^6)
Time: 0.800
> time LTaylor(L, 1, 5);
1.87710082755801E-24*$.1 - 3.47478614210524E-24*$.1^2 + 1.73184990011930*$.1^3
   - 3.20590558844390*$.1^4 + 2.80009237167013*$.1^5 + O($.1^6)
Time: 1.530

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And this is the leading derivative, with the same value as Evaluate(L,1:D:=3).

> c := Coefficient($1,3)*Factorial(3);c;
10.3910994007158

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Finally, we compute the 3rd derivative of the modified \(L\)-function \(L^*(s)=\gamma(s)L(s)\) at \(s=1\). For an elliptic curve over the rationals, \(\gamma(s)=(N/\pi^2)^{s/2}\Gamma(s/2)\Gamma((s+1)/2)\), where \(N\) is the conductor. So, by the chain rule, \({L^*}'''(1)=\gamma(1)L'''(1)=\sqrt{N/\pi}L'''(1)\).

> LStar(L, 1 : Derivative:=3);
417.724689268266
> c*Sqrt(Conductor(E)/Pi(RealField(15)));
417.724689268267

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DedekindZetaExact(K, z): Fld, RngIntElt -> FldRatElt

Relative: BoolElt                    Default: false

The special value of the Dedekind zeta function of \(K\) at the negative integer \(z\). Returns \(\zeta_K(z)\) or \({\zeta_K(z) \over \zeta_F(z)} = L(\chi_{K/F}, z)\) if \(K\) is a quadratic extension of \(F\) and Relative is true. In the latter case, can also compute the value at \(z = 0\).