Order of Zeros of Dedekind Zeta Functions

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity at least 2 if $L$ has a subfield $K$ for which $L/K$ is a nonabelian Galois extension. We also extend this to zeros of order 3 when $\operatorname{Gal}(L/K)$ has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.