$F$, we can associate a well-defined Frobenius conjugacy class ${\mathrm{Fr}}_v$ of $\operatorname{Gal}(\bar F/F)$, and $L(s, \rho_v)$ determines the eigenvalues of $\rho({\mathrm{Fr}}_v)$, and thus $\operatorname{tr}\rho({\mathrm{Fr}}_v)$. Let $S$ be a set of places of $F$, and suppose $L(s, \rho_v) = L(s, \rho'_v)$, or even just $\operatorname{tr}\rho({\mathrm{Fr}}_v) = \operatorname{tr}\rho'({\mathrm{Fr}}_v)$, for all $v \not \in S$.

Continuity means that $\rho$ and $\rho'$ factor through a common finite quotient $G = \operatorname{Gal}(K/F)$ of $\operatorname{Gal}(\bar F/F)$, for some finite normal extension $K/F$. View $\rho, \rho'$ as irreducible $n$-dimensional representations of the finite group $G$. The Chebotarev density theorem tells us that if $C$ is a conjugacy class in $G$, then the image of ${\mathrm{Fr}}_v$ in $\operatorname{Gal}(K/F)$ lies in $C$ for a set of primes $v$ of density $\frac{|C|}{|G|}$. This implies that if the density of $S$ is $< \delta_n(G)$ (or $< \delta_n^\natural(G)$ if $\rho, \rho'$ are primitive), then $\rho \simeq \rho'$, i.e., $L(s, \rho) = L(s, \rho')$. Moreover, this bound on the density of $S$ is sharp.

Consequently, Proposition \[prop1\] tells us that if the density of $S$ is $< \frac 1{2n^2}$, then $L(s, \rho) = L(s, \rho')$, and Serre’s result implies this bound is sharp. (See Rajan [@rajan] for an analogous result on $\ell$-adic Galois 