about (2), which is our primary motivation, though both are natural questions. Our main results about the first question are for comparing primitive characters of degree $\le 3$ and characters of $\operatorname{PSL}(2,q)$ or $\operatorname{SL}(2,q)$. This will yield sharp bounds on how many Euler factors one needs to distinguish primitive 2- or 3-dimensional $L$-functions of Galois representations. We address them in turn.

Distinguishing group characters
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Let $G$ be a finite group, and $\rho, \rho'$ be two complex representations of $G$ with characters $\chi, \chi'$. We will study the quantities $$\delta(\rho, \rho') = \delta(\chi, \chi') = 
 \frac  {| \{ g \in G : \chi(g) \ne \chi'(g) \} |}{|G|}.$$

Specifically, let $\delta_n(G)$ be the minimum of $\delta(\rho, \rho')$ as $\rho, \rho'$ range over pairs of inequivalent irreducible $n$-dimensional representations of $G$, with the convention that $\delta_n(G) = 1$ if there are no such pairs $\rho, \rho'$. Note that $\delta_n(G)$ tells us what fraction of elements of $G$ we must check to distinguish irreducible degree $n$ characters. Put $d_n = \inf_G \{ \delta_n(G) \}$.

An elementary consequence of orthogonality relations is

\[prop1\] We have $d_n \ge  \frac 1{2n^2}$.

Buzzard, Edixhoven and Taylor constructed examples to show this bound is sharp when $n$ is 