= Solution
There is an absolute $c>0$ such that the product of the <Dirichlet L-functions> modulo $q$ has no zero in
$$
\boxed{\sigma\geq1-\frac{c}{\log(q(|t|+2))}}
$$
except possibly one zero. If it exists, this <exceptional zero> $\beta$ is real and simple, belongs to a real nonprincipal character $\chi_1$, and lies very close to one. Among the primitive characters whose conductors divide $q$, at most one can have such a zero. This is the <Classical zero-free region for Dirichlet L-functions>.
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