A Fast Algorithm to Compute l(1/2, f x \chi_q)

TL;DR

This paper introduces a fast algorithm for computing the central value of the L-function associated with a modular cusp form twisted by a Dirichlet character, especially efficient for smooth or highly composite moduli.

Contribution

The paper presents a novel algorithm that computes L(1/2, f × χ_q) efficiently, with complexity depending on the smoothness of q, improving computational methods for these special values.

Findings

Algorithm computes L-values up to desired precision.

Time complexity is O(1 + |q|^{5/6+o(1)}) for smooth or composite q.

Applicable to holomorphic or Maass cusp forms.

Abstract

Let $f$ be a fixed (holomorphic or Maass) modular cusp form. Let $\cq$ be a Dirichlet character mod $q$. We describe a fast algorithm that computes the value $L(1/2,f\times\chi_q)$ up to any specified precision. In the case when $q$ is smooth or highly composite integer, the time complexity of the algorithm is given by $O(1+|q|^{5/6+o(1)})$.