---
abstract: 'Serre obtained a sharp bound on how often two irreducible degree $n$ complex characters of a finite group can agree, which tells us how many local factors determine an Artin $L$-function. We consider the more delicate question of finding a sharp bound when these objects are primitive, and answer these questions for $n=2,3$. This provides some insight on refined strong multiplicity one phenomena for automorphic representations of $\operatorname{GL}(n)$. For general $n$, we also answer the character question for the families $\operatorname{PSL}(2,q)$ and $\operatorname{SL}(2,q)$.'
address:
- 'Department of Mathematics, University of Oklahoma, Norman, OK 73019 USA'
- 'Department of Mathematics, Occidental College, Los Angeles, CA 90041 USA'
author:
- Kimball Martin
- Nahid Walji
title: 'Distinguishing finite group characters and refined local-global phenomena'
---

Introduction
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In this paper, we consider two questions about seemingly different topics:

1.  How often can two characters of a finite group agree?

2.  How many local Euler factors determine an $L$-function?

The first question is just about characters of finite groups, and the second is a refined local-global principle in number theory. However, it has been observed, notably by Serre, that being able to say something about (1) allows one to say something 