The least prime with a given cycle type

Let $G$ be a finite group. Let $K\mathrm{/}k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C\subseteq \mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}k)\simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\mathrm{N}\mathfrak{p}\ll \mathrm{\mid }\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{c}(K){\mathrm{\mid }}^{\alpha }$ for some constant $\alpha =\alpha (G,C)$. There is a rich literature establishing unconditional admissible values for $\alpha$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $\alpha$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2>0$ such that for any $n\ge 2$ and any conjugacy class $C\subset S_n$, one may take $\alpha (S_n,C)=c_1\exp (-c_{2}n)$. Our approach reduces the core problem to a question in character theory.