The Riemann zeta function is defined as: $$ \zeta(s) := \sum_{n=1}^{\infty}\frac{1}{n^s} $$ for all $s$ in the $\textrm{Re}(s)>1$ half-plane.

In order to distinguish between $\sum_{n=1}^{\infty}\frac{1}{n^s}$ and the Euler product expression of $\zeta$, I usually refer to $\sum_{n=1}^{\infty}\frac{1}{n^s}$ as the Dirichlet series expression of $\zeta$.

The same applies for Dirichlet $L$-series: $$ L(\chi,s) = \sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}. $$ $\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ is the Dirichlet series expression of the Dirichlet $L$-series $L(\chi,s)$ (again, as distinct from the Euler product expression).

Now we come the the Dedekind zeta function, defined in terms of some algebraic number field $K$: $$ \zeta_K(s) := \sum_{\mathfrak{a}}\frac{1}{N(\mathfrak{a})^s}, $$ where the sum is taken over all integral ideals $\mathfrak{a} \subset \mathcal{O}_K$, and $N$ denotes the ideal norm in $\mathcal{O}_K$.

My question is: What do we call $\sum_{\mathfrak{a}}\frac{1}{N(\mathfrak{a})^s}$? Note that it is not a Dirichlet series.

Might we call it the additive series expression (again, as distinct from the Euler product expression)?

Any suggestions, or does anyone know of a universally accepted standard name for it?

Many thanks.