Joint extreme values of L-functions on and off the critical line

It is shown that any number of distinct primitive GL(1) and GL(2) L-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such L-functions. The joint distribution of GL(m) L-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet L-functions and generally improve upon the work of Mahatab, Pańkowski and Vatwani on the class of L-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, is the resonance method of Soundararajan and Hilberdink/Voronin, respectively. On the critical line we additionally introduce a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function which makes it possible to avoid using any information on the zero distribution of L-functions whose degree is less than three.