Discrete value-distribution of Artin $L$-functions associated with cubic fields

Abstract

Arising from the factorizations of Dedekind zeta-functions associated with non-Galois cubic fields, we obtain Artin L-functions of two-dimensional representations.In this paper, we study the value-distribution of such Artin L-functions in the aspect of discriminants of cubic fields. We prove that averages of values of the Artin L-functions are expressed by integrals involving a density function which can be explicitly described. As a corollary, we obtain asymptotic formulas of counting functions for certain families of cubic fields.