For a lattice $\Lambda$ of rank $n$ in $\mathbb{Q}^{n\times n}$ whose non-zero elements are inversible matrices, let $$Z(s,\Lambda) = \sum_{a \in \Lambda^*} |\det(a)|^{-s}$$
**I wonder if (and how to show) this Dirichlet series has a functional equation ?**

The motivation is that $\zeta_K(s)$ the Dedekind zeta function of a number field is of the form $\zeta_K(s) =\frac{1}{|\mathcal{O}_K^\times|}\sum_{J \in C_K} N(J)^{s} Z(s,\Lambda_J)$ where $C_K$ is the ideal class group, $\mathcal{O}_K$ being isomorphic to such a lattice in $\mathbb{Q}^{n \times n}$.

Aurel commented that for most number fields $\sum_{a \in \mathcal{O}_K^*} |N(a)|^{-s}$ diverges (infinitely many units), so we have to look at $\sum_{a \in \mathcal{O}_K^*/\mathcal{O}_K^\times} |N(a)|^{-s}$, and on the lattice side, except when $\Lambda$ is (up to a constant) a subring of $\mathbb{Q}^{n\times n}$, it is not obvious how to translate it.

9more comments