= Solution
Let $G=\operatorname{Cl}(K)$ and let $\widehat G$ be its character group. Character orthogonality gives
$$
\mathbf1_{[\mathfrak p]=C}
=\frac1{h_K}\sum_{\chi\in\widehat G}
\overline{\chi(C)}\,\chi([\mathfrak p]).
$$
For $\operatorname{Re}s>1$, the prime term of $\log L(s,\chi)$ is
$$
\sum_{\mathfrak p}\chi([\mathfrak p])N\mathfrak p^{-s},
$$
up to a function bounded as $s\to1^+$, since higher prime powers converge there. The trivial character contributes
$$
\log\frac1{s-1}+O(1),
$$
whereas every nontrivial character contributes $O(1)$ because its $L$-function is nonzero at $1$. Therefore
$$
\sum_{[\mathfrak p]=C}N\mathfrak p^{-s}
=\frac1{h_K}\log\frac1{s-1}+O(1),
$$
and the density is $\boxed{1/h_K}$.
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