The average of the smallest prime in a conjugacy class

Let $C$ be a conjugacy class of $S_n$ and $K$ an $S_n$-field. Let $n_{K,C}$ be the smallest prime which is ramified or whose Frobenius automorphism Frob$_p$ does not belong to $C$. Under some technical conjectures, we compute the average of $n_{K,C}$. For $S_3$ and $S_4$-fields, our result is unconditional. For $S_n$-fields, $n=3,4,5$, we give a different proof which depends on the strong Artin conjecture. Let $N_{K,C}$ be the smallest prime for which Frob$_p$ belongs to $C$. For $S_3$-fields, we obtain an unconditional result for the average of $N_{K,C}$ for $C=[(12)]$.

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