Computing L-Functions of Quadratic Characters at Negative Integers

TL;DR

This paper reviews various computational methods for evaluating L-functions of quadratic characters at negative integers, highlighting the most effective techniques depending on the size of the integer and the conductor.

Contribution

It provides a comparative analysis of methods for computing L-values of quadratic characters, emphasizing the practical efficiency of Eisenstein series and functional equations.

Findings

Eisenstein series are optimal for small k (up to 100).

Functional equations are best for large k unless the conductor is very large.

Method choice depends on the size of k and the conductor of the character.

Abstract

We survey a number of different methods for computing $L(\chi,1-k)$ for a Dirichlet character $\chi$, with particular emphasis on quadratic characters. The main conclusion is that when $k$ is not too large (for instance $k\le100$) the best method comes from the use of Eisenstein series of half-integral weight, while when $k$ is large the best method is the use of the complete functional equation, unless the conductor of $\chi$ is really large, in which case the previous method again prevails.