For unconditional results, see Theorem 11.4 of Montgomery-Vaughan, which states the following. Let $\chi$ be a primitive Dirichlet character modulo $q > 1$. Then there exists a constant $c > 0$ such that $L(s,\chi)$ has at most one zero in the region
$$\Re(s) > 1 - \frac{c}{\log(q(|\Im(s)| + 3))},$$
and this exceptional zero can only exist if $\chi$ is quadratic, in which case this zero is real. (This is the standard zero-free region for Dirichlet $L$-functions.) We denote by $\beta$ such an exceptional zero (which is called a Landau-Siegel zero). Then if $s$ is in this region and additionally if $|s - \beta| > 1/\log q$ if $\beta$ exists, we have that
$$\frac{1}{L(s,\chi)} \ll \log(q(|\Im(s)| + 3)).$$
If $\beta$ exists, $s$ is in this region, and additionally if $0 < |s - \beta| \leq 1/\log q$, then
$$\frac{1}{|s - \beta| (\log q)^2} \ll \frac{1}{|L(s,\chi)|} \ll \frac{1}{|s - \beta|}.$$