Overview#

A large variety of \(\zeta\)-functions and \(L\)-functions occur in number theory and algebraic geometry. Some well-known \(L\)-functions include the Riemann \(\zeta\)-function, the Dedekind \(\zeta\)-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

\[E: y^2+xy=x^3-x^2\]

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 \(\zeta\)-function, the Dedekind \(\zeta\)-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 \(s_0\) 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 [Shafarevich, 1995] Chapter 4, Serre [Serre, 1965] and articles in [Jannsen et al., 1994] for a background on \(L\)-functions. The algorithms mostly follow Dokchitser [Dokchitser, 2004] and the Pari implementation ComputeL [Dokchitser, 2002]. See also Lavrik [Lavrik, 1967], Tollis [Tollis, 1997] and the exposition in Cohen [Cohen, 2000], 10.3.