Classical zero-free region for Dirichlet L-functions

There is an absolute $c>0$ such that the Dirichlet L-functions modulo $q$ have no zero inapart from at most one real simple zero belonging to a real nonprincipal character.

An exceptional zero is the possible real simple zero $\beta$ omitted from the otherwise uniform Classical zero-free region for Dirichlet L-functions. It belongs to a real nonprincipal primitive Dirichlet character and satisfies $\beta <1$.

Uniformly for $x\ge 2$ and $(a,q)=1$,where the second term occurs only when a real character ${\chi }_1$ has an exceptional zero $\beta$.

If some $ϵ>0$ satisfiesuniformly for sufficiently large $q$, then every exceptional zero modulo $q$ satisfiesChoose ${\chi }_1(a)=-1$ and $x=\exp (A(\log q{)}^2)$ in the exceptional-zero asymptotic, with $A$ large in terms of $ϵ$.