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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 $1$?

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    $\begingroup$ It means that this $c$ applies for all Dirichlet Characters of all moduli, provided their product isn't identically $1$; in particular, if you have an exceptional zero for one modulus $q$ ($\beta > 1-\frac{b}{\log q}$, $b$ small absolute constant), the next exceptional zero $\beta_1$ cannot appear until a modulus $q_1$ which is considerably bigger than $q$, so the original $\beta > 1-\frac{b}{\log q}$ satisfies also $\beta \le 1-\frac{c}{\log {(qq_1)}}$ hence if you calibrate $b,c$ you get various estimates for how big $q_1$ must be w.r. to $q$ $\endgroup$ – Conrad Apr 22 at 21:24

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