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

It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-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 $\mathrm{GL}(m)$ $L$-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, Pa\'nkowski 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.