# Epstein zeta function for non-fundamental discriminant to L-series

Let $$Q(x,y) = ax^2+b xy + cy^2$$ be a primitive integral positive-definite quadratic form, with associated number field $$K$$. If $$D=b^2-4ac$$ is a fundamental discriminant, then it's well-known that $$\zeta_Q(s) = \sum_{(x,y)\in \mathbb{Z}^2-0} \frac{1}{Q(x,y)^s}$$ is essentially partial zeta series $$\zeta_K(I,s)$$ for an ideal class of $$K$$, so $$\zeta_Q(s)$$ is a linear combination of Artin L-function for characters of $$\text{Gal}(L/K)$$, here $$L$$ is Hilbert class field of $$K$$.

If $$D$$ is not fundamental, let $$\mathcal{O}$$ be the associated non-maximal order. I think $$\zeta_Q(s)$$ is still of the following form $$\tag{*}\zeta_Q(s) = \sum_i f_i(s) L(\chi_i,s)$$ here $$f_i(s)$$ is a certain finite Euler product, and $$L(\chi_i,s)$$ are Artin L-function with $$\chi_i$$ character of $$\text{Gal}(L/K)$$, with $$L$$ the ring class field of $$\mathcal{O}$$.

Question: is $$(*)$$ true in general? how to explicitly find the $$f_i(s)$$?

I think an explicit form of $$f_i$$ would likely to be complicated, so proving their existence is already nice enough.

As an example of $$(*)$$: let $$Q(x,y) = x^2 + 4y^2, K = \mathbb{Q}(\sqrt{-1}), \mathcal{O} = \mathbb{Z}[2\sqrt{-1}]$$, ring class field $$L$$ is same as $$K$$, so there is only one $$L(\chi_i,s) = \zeta_K(s)$$, we have $$\sum_{(x,y)\in \mathbb{Z}^2-0} \frac{1}{(x^2+4y^2)^s} = 2 \left(1-2^{-s}+2\times 4^{-s}\right) \zeta_K(s)$$

K. Williams and others theorem 10.1 of this paper have proved that $$(*)$$ is true if the class group of $$\mathcal{O}$$ is $$2$$-torsion; more generally, $$(*)$$ is true if we sum over a genus.

Any idea is welcomed, thank you.

• Please use a high-level tag like "nt.number-theory". I added this tag now. Commented May 25, 2023 at 16:17