L-functions of curves of genus ≥ 3

Abstract

We introduce an algorithm (written in Sage) for the L-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local L-factor and the conductor exponent at the primes of bad reduction. The method we use works for any superelliptic curve over a number field. We present several families of examples, e.g. hyperelliptic curves of genus 2,...,6 with semistable reduction everywhere. In a second part, we consider Picard curves over Q. We give a completed list of the curves resulting from the algorithm by Malmskog and Rasmussen. Moreover, we present generalizations of their approach. This leads to a recipe for the calculation of the smallest conductor of a Picard curve over Q.