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

## TL;DR

This paper investigates the distribution of Artin L-functions linked to cubic fields, revealing their mean values and implications for class number distribution using explicit density functions.

## Contribution

It introduces a new analysis of Artin L-functions for non-Galois cubic fields, connecting their value distribution to class number statistics in conductor aspect.

## Key findings

- Mean values of Artin L-functions are expressed via explicit integrals.
- Distribution of class numbers of cubic fields is analyzed.
- Density functions for L-value distributions are constructed explicitly.

## 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.