The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect

Abstract

Arising from the factorizations of Dedekind zeta-functions of cubic fields, we obtain Artin $L$-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin $L$-functions for families of non-Galois cubic fields in conductor aspect. We prove that various mean values of the Artin $L$-functions are represented by integrals involving a density function which can be explicitly constructed. By the class number formula, the result is applied to the study on the distribution of class numbers of cubic fields.

50 pages

References in corpus (3)

• Secondary terms in counting functions for cubic fields
• Distribution of values of $L$-functions at the edge of the critical strip
• Probability density functions attached to random Euler products for automorphic $L$-functions