When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, it is also known that the Riemann $\zeta$ function does not vanish on (0, 1).

Are there any result showing that some Dirichlet L-functions (attached to a non-principal character) do not vanish *far* from $1$ ? I think non-vanishing at $1/2$ is still open in general, but maybe it is known in some cases.