Suppose $\chi_1$ and $\chi_2$ are Dirichlet characters moduli $q_1$ and $q_2$ and suppose they are induced by same primitive character. If $p$ is a prime greater than both $q_1$ and $q_2$ then clearly $\chi_1(p)=\chi_2(p).$
My question is about the converse.
Suppose $\chi_1 (p) = \chi_2(p)$ for all primes except finitely many, I suspect that $\chi_1 $ and $\chi_2$ are induced by same primitive character. How to prove or disprove this statement?