Finding p-adic zeroes of the Kubota-Leopoldt zeta-function numerically

Abstract

We first establish why the p-adic zeta function has a Dirichlet series expansion. We then compute an improved expansion, which allows us to express it as a power-series modulo pⁿ. Using this expansion, we compute all the zeros of L₋p(s, χωʲ) for those quadratic characters χ of conductor < 200.