p-Adic Zeroes of the Kubota-Leopoldt Zeta-Function

Abstract

In this paper we establish why the p-adic zeta function has a Dirichlet series expansion. We 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, χω^j) for those quadratic characters χ of conductor < 200. For the calculation we use a PARI-GP Program.