representations.) In fact, this application to Galois representations was Serre’s motivation, and it also motivated the bound in Ramakrishnan’s conjecture. For us, the Chebotarev density theorem together with Theorem \[thm1\] yields

Let $\rho$, $\rho'$ be irreducible primitive $n$-dimensional Artin representations for $F$. Suppose $\operatorname{tr}\rho({\mathrm{Fr}}_v) = \operatorname{tr}\rho'({\mathrm{Fr}}_v)$ for a set of primes $v$ of $F$ of density $c$.

1.  If $n=2$ and $c > \frac 34$, then $\rho \simeq \rho'$.

2.  If $n=3$ and $c > \frac 57$, then $\rho \simeq \rho'$.

When $n=2$, if $\rho$ and $\rho'$ are automorphic, i.e., satisfy the strong Artin conjecture, then the above result already follows by [@walji]. When $n=2$, the strong Artin conjecture for $\rho$ is known in many cases—for instance, if $\rho$ has solvable image by Langlands [@langlands] and Tunnell [@tunnell], or if $F={\mathbb Q}$ and $\rho$ is “odd” via Serre’s conjecture by Khare-Wintenberger [@khare-wintenberger]. We remark that the methods of [@walji] are quite different than ours here.

The above corollary suggests the following statement may be true: if $\pi, \pi'$ are cuspidal automorphic representations of $\operatorname{GL}_3({\mathbb A}_F)$ which are not induced from characters and $L(s, \pi_v) = L(s, \pi'_v)$ for a set of primes $v$ of density $> \frac 57$, then $\pi 