I often read about analytic number theory on $\mathbf{Q}$ when it turns to explicit computations, so I wonder how much results generalize as they stand and the use of $\mathbf{Q}$ is merely made for ease of notations. Let $F$ be a number field, $\mathfrak{p}$ a prime ideal and $\mathfrak{n}$ denotes generically an integer ideal.
Does the decomposition of $L$-functions in Euler product still holds: $$L(s, f) = \sum_{\mathfrak{n}} \frac{c_\pi(\mathfrak{n})}{N(\mathfrak{n})^s} = \prod_\mathfrak{p} \prod_{j=1}^n (1-\alpha_j(\mathfrak{p})N(\mathfrak{p})^{-s})^{-1} \quad ?$$
And does the Hecke theory and Hecke operators works as well, giving the coefficients $c_\pi(\mathfrak{n})$ up to normalization, etc. with for instance $$T_\mathfrak{p} = \left( \begin{array}{cc} \mathfrak{p} & \\ & 1 \end{array} \right) \quad ?$$
Is there a reference for these general notations?
