What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?

Question

Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean the number of ideals of norm $n$, that belong to the class $[c]$,

$$r(n)=r([c], n)= \sharp\bigg\{ \mathfrak{I} \subseteq \mathcal{O}_L: \mathfrak{I} \in [c], N(\mathfrak{I})=n \bigg\}.$$

Dedekind zeta function, is something which is related to all of these $r([c], n)$'s, where $[c]$ varies arbitrarily in $\mathcal{Cl(O}_L)$. I am curious about the situation when we restrict ourselves to only one arbitrary but fixed class $[c] \in \mathcal{Cl(O}_L)$: What are the "well-known" functions in number theory, that are related to $r(n)=r([c], n)$'s?

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What I know: Suppose that $L$ is an imaginary quadratic field, $D$ its discriminant, and let $[c]$ be an arbitrary but fixed class in $\mathcal{Cl(O}_L)$, and let $\Theta(z)=\Theta_{[c]}(z)= 1+\sum_{n=1}^{\infty} r(n)q^n$, where $q=e^{2\pi i z}$, and $r(n)=r([c], n)$. Then $\Theta(z)$ is a modular form of weight $1$, with level $N=\vert D \vert$ and charachter $\chi(.)=\left(\dfrac{D}{.}\right)$; i.e. for $z \in \mathcal{H}$ and $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$ we have $\Theta\left(\dfrac{az+b}{cz+d}\right) = \chi(a)\left(cz+d\right) \Theta(z)$. [We can associate a quadratic form the class $[c]$, and here $\Theta(z)$ is the associated theta series.]

Answer 1

It sounds like you're looking for something like the function $$\zeta_C(s) = \sum_{\mathfrak{a} \in C} N(\mathfrak{a})^{-s} = \sum_{n \ge 1} r([c], n) n^{-s}.$$ These functions are sometimes called "ideal class zeta functions" and they come up from time to time in the literature. See e.g. this paper in J London Math Soc:

Friedman, Eduardo, The zero near 1 of an ideal class zeta function, J. Lond. Math. Soc., II. Ser. 35, 1-17 (1987). ZBL0583.12010.

The related functions where you sum over a character $\psi$ of the ideal class group, $$L(\psi, s) = \sum_{C \in Cl(L)} \psi(C) \zeta_C(s),$$ are much more commonly studied (since they have an Euler product expansion, which isn't true of the $\zeta_C(s)$ individually). These are examples of Hecke $L$-functions.