  In the present paper, we study large values of Dirichlet $L$- functions
inside the critical strip. For every $1/2<\sigma<1$, 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 $\Re(s)=1$. We show how to
adapt the argument to our setting.
