In my analytic number theory course, we discussed Dirichlet Series of Dirichlet Characters. A question I was asked was to show that:
If $\chi_1, \chi_2$ are two Dirichlet Characters of modulo $q_1, q_2$ respectively such that $\chi_1\chi_2 \not \equiv 1$, then:
$L(s,\chi_1)L(s, \chi_2)$ has at most one real zero, $\beta$ such that $1 > \beta > 1 - \frac{c}{log(q_1q_2)}$, where $c>0$ is some absolute constant.
What does "absolute" mean here? Does it mean that this $c$ applies for all Dirichlet Characters of all moduli, provided their product isn't identically $1$? Does it mean that it only works specifically for the moduli $q_1, q_2$ but that it'll work for any of the Dirichlet Characters modulo $q_1, q_2$ such that the product is not identically $1$?