  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 GL$(n)$. For
general $n$, we also answer the character question for the families PSL$(2,q)$
and SL$(2,q)$.
