Large values of Dirichlet $L$- functions inside the critical strip

In the present paper, we study large values of Dirichlet $L$- functions inside the critical strip. For every $1\mathrm{/}2<\sigma <1$, we show that for $q$ sufficiently large, there exists a non-principal character $\chi$ modulo $q$ and a constant $c(\sigma )>0$ such that $\log \vert L(\sigma,\chi)\vert \gg c(\sigma)(\log q)^{1-\sigma}(\log\log q)^{-\sigma}$. This matches the believed prediction for these values which was previously known only for the Riemann zeta function since Montgomery, or conditionally on GRH for quadratic $L$- functions due to Lamzouri. In a recent work involving the author, a new implementation of the resonance method was presented in order to exhibit large values of the Riemann zeta function on the line $\mathrm{\Re }(s)=1$. We show how to adapt the argument to our setting.