On large values of L($\upsigma$,$\upchi$)

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 is an element of (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 is an element of (1/2, 1), we show that for all sufficiently large q, there is a non-principal character chi(mod q) such that log vertical bar L (sigma, chi)vertical bar >= C (s)(log q)(1-sigma) (loglog q)(-sigma). In the case sigma = 1, we show that there is a nonprincipal character chi(mod q) for which vertical bar L (1, chi)vertical bar >= e(gamma)(log(2)q + log(3)q - C). In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.