Artin L-functions of small conductor

TL;DR

This paper investigates the minimal conductors of Artin L-functions for specific Galois types, providing improved bounds and exact values for small types through analytic methods and number field tables.

Contribution

It introduces adapted analytic techniques for fixed Galois types and determines the smallest conductors for small types using comprehensive number field data.

Findings

Improved lower bounds on smallest conductors for Artin L-functions.

Exact smallest conductors identified for small Galois types.

Enhanced understanding of the relationship between Galois types and conductors.

Abstract

We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and get much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.