Sharp conditional moment bounds for products of $L$-functions

Assuming the Generalized Riemann Hypothesis and the Generalized Ramanujan Conjecture, we determine the order of the $2(k_1,\dots ,k_r)$th moment of a product of distinct irreducible $L$-functions on the critical line. As a consequence, we obtain conditional information about the independence of these $L$-functions in the large deviations regime. We also obtain sharp moment bounds for Hurwitz zeta functions with rational parameter, and a certain family of Dedekind zeta functions.