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(s0) for complex s0, values for the derivatives L(k)(s0) 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 s0, the intrinsic computes either L(s0), or if D>0, the value of the derivative L(D)(s0). If D>0 and it is known that all the lower derivatives vanish, L(s0)=L'(s0)=...=L(D - 1)(s0)=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 s0 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 s0, the intrinsic computes either the value L * (s0) or, if D>0, the value of the derivative L * (D)(s0). Here L * (s)=γ(s)L(s) is the modified L-function that satisfies the functional equation (cf. Section Terminology)

L * (s) = (sign) .bar L * ((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=s0, where s0 is a complex number:

L(s0) + L'(s0)x + L"(s0)x2/2! + ... + L(n)(s0)xn/n! + O(xn + 1) .

If the first few terms L(s0), ..., L(k)(s0) of this expansion are known to be zero, the computation time can be reduced by setting ZeroBelow:=k+1.

Example Lseries_lseries-evaluate (H139E19)

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

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

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

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
Finally, we compute the 3rd derivative of the modified L-function L * (s)=γ(s)L(s) at s=1. For an elliptic curve over the rationals, γ(s)=(N/π2)s/2Γ(s/2)Γ((s + 1)/2), where N is the conductor. So, by the chain rule, (L * )"'(1)=γ(1)L"'(1)=Sqrt(N/π)L"'(1). 

> LStar(L, 1 : Derivative:=3);
417.724689268266
> c*Sqrt(Conductor(E)/Pi(RealField(15)));
417.724689268267
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 ζK(z) or (ζK(z) /ζF(z)) = L(χK/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.
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Version: V2.29 of Fri Nov 28 15:14:01 AEDT 2025
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