Let \(K\) be a number field. Define, for a fractional ideal \(\mathcal{I} \subseteq K\), define \[\begin{equation} \zeta_{[\mathcal{I}]}(s) = \mathop{\mathrm{N}}(\mathcal{I})^{s} \sum_{v \in \mathcal{I}\setminus \{0\}} |\mathop{\mathrm{N}}(v)|^{-s}. \end{equation}\] Observe that \(\zeta_{[\mathcal{I}]}(s)\) depends only on the ideal class of \(\mathcal{I}\) in the class group \(\mathop{\mathrm{Cl}}(K)\) of \(K\). Furthermore, observe that the Dedekind zeta function \(\zeta_{K}(s)\) satisfies \[\begin{equation} \zeta_{K}(s) = \sum_{[\mathcal{I}] \in \mathop{\mathrm{Cl}}(K)} \zeta_{[\mathcal{I}]}(s). \end{equation}\]

In general, this can also be used to define class group \(L\)-functions of a number fields as \[\begin{equation} L(s,\chi) = \sum_{[\mathcal{I}] \in \mathop{\mathrm{Cl}}(K)} \zeta_{[\mathcal{I}]}(s) \chi([\mathcal{I}]), \end{equation}\] for a character \(\chi:\mathop{\mathrm{Cl}}(K) \rightarrow \mathbb{C}\).

[1] is a pretty good introduction for these functions. They discuss a non-vanishing theorem for these L-functions.

References