On large values of $L(\sigma,\chi)$

In recent years a variant of the resonance method was developed which allowed to obtain improved $\Omega$-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be adapted to prove the existence of large values of $|L(\sigma, \chi)|$ in the range $\sigma \in (1/2,1]$, and to estimate the proportion of characters for which $|L(\sigma, \chi)|$ is of such a large order. More precisely, for every fixed $\sigma \in (1/2,1)$ we show that for all sufficiently large $q$ there is a non-principal character $\chi$ (mod $q$) such that $\log |L(\sigma,\chi)| \geq C(\sigma) (\log q)^{1-\sigma} (\log \log q)^{-\sigma}$. In the case $\sigma=1$ we show that there is a non-principal character $\chi$ (mod $q$) for which $|L(1,\chi)| \geq e^\gamma \left(\log_2 q + \log_3 q - C \right)$. In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.