Sharp lower bounds for shifted moments of Dedekind zeta functions

Let $K_1,\cdots ,K_r$ be fixed number fields, and let $L$ be the compositum of their Galois closures. Assuming GRH for ${\zeta }_L$, we prove a sharp lower bound for products of shifted Dedekind zeta functions on the critical line, for arbitrary fixed positive real exponents and uniformly for shifts of size at most $T\mathrm{/}2$. The correlation factor is expressed as a product of Dedekind zeta functions of the fixed fields of double-coset stabilisers in $\text{Gal}(L\mathrm{/}\mathbb{Q})$. Combined with the corresponding upper bound by the authors, determines the order of magnitude of these shifted moments for both Galois and non-Galois fields.