A large variety of ζ-functions and L-functions occur in number
theory and algebraic geometry. Some well-known L-functions include
the Riemann ζ-function, the Dedekind ζ-function of a number
field, Dirichlet series associated to characters, and L-series
of curves (e.g., elliptic curves) over the rationals. Magma provides
functionality for constructing such L-functions and computing their
values in the complex plane. A typical calculation might go as follows:
> L := LSeries(EllipticCurve([0, -1, 1, 0, 0]));
> Evaluate(L,2);
0.546048036215013518334126660433
The first line defines an L-series L(E, s) of the elliptic curve
E: y2 + xy=x3 - x2
while the second line computes its value at s=2. An impatient reader
may wish simply to type LSeries; at the prompt and look at the
various LSeries signatures and mimic the code above, thus getting
access to much of the functionality.
Topics covered in this chapter include:
- The built-in L-series which include the Riemann
ζ-function, the Dedekind ζ-function of a number field,
Dirichlet series associated to characters, Artin representations,
modular forms, and L-series of elliptic curves;
- The calculation of values, derivatives and Taylor
expansions of L-series at a complex point s0 to desired accuracy;
- A technical description of the L-series object in
Magma, together with a description of how to construct user-defined
L-series with any number of gamma factors, provided that the L-series
satisfies a functional equation of the standard type;
- Operations such as division, multiplication and the
tensor product of two L-series.
The reader is referred to Manin-Panchishkin
[Sha95] Chapter 4,
Serre
[Ser65] and articles in
[JKS94] for a background
on L-functions. The algorithms mostly follow Dokchitser
[Dok04]
and the Pari implementation ComputeL
[Dok02]. See also
Lavrik
[Lav67], Tollis
[Tol97] and the exposition
in Cohen
[Coh00], 10.3.
V2.28, 13 July 2023