Distinguishing finite group characters and refined local-global phenomena

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$(2,q)$.