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.



Real non trivial zeros of Dirichlet L-functions
Perhaps the most comprehensive result in this direction is due to Conrey and Soundararajan (<https://arxiv.org/abs/math/0111013> or <https://link.springer.com/article/10.1007/s00222-002-0227-x>). They prove (among other things) that the density of odd positive squarefree integers $d\leq x$ such that $L(s,\chi\_{8d})>0$ for all $s\in[0,1]$ is at least $\frac{4}{5\pi^2}$. For work on nonvanishing at $s=1/2$, Soundararajan (<https://arxiv.org/abs/math/9902163> or <https://www.jstor.org/stable/2661390?seq=1#page_scan_tab_contents>) obtains more robust results.


