= Solution
By the <Kronecker–Weber theorem>, choose $m$ with
$$
K\subseteq\mathbb Q(\zeta_m).
$$
Restriction gives a quotient
$$
(\mathbb Z/m\mathbb Z)^\times
\twoheadrightarrow G=\operatorname{Gal}(K/\mathbb Q).
$$
Every character $\psi\in\widehat G$ inflates along this quotient and has an associated primitive Dirichlet character $\chi_\psi$, whose conductor may divide $m$. Comparing Euler factors, or applying the factorization of the Artin $L$-function of the regular representation, gives
$$
\boxed{\zeta_K(s)=
\prod_{\psi\in\widehat G}L(s,\chi_\psi).}
$$
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