Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the characteristic at $\mathfrak{p}$, there is an action of $\hat{\mathbb{Z}}$ on the inertia invariants $H^m_{et}(X_{\bar{k}},\mathbb{Q}_l)^I$ and we can define \begin{equation} L_{\mathfrak{p}}(H^m(X),s)=\text{det}(1-f_{\mathfrak{p}}\cdot N\mathfrak{p}^{-s}|H^m_{et}(X_{\bar{k}},\mathbb{Q}_l)^I), \end{equation} where $f_{\mathfrak{p}}\in\hat{\mathbb{Z}}$ is the geometric Frobenius. In some half plane, The Euler product over all $\mathfrak{p}$ is the mth Hasse-Weil zeta function: \begin{equation} L(H^m(X),s)=\prod_{\mathfrak{p}}(H^m(X),s). \end{equation} Their alternating product is the Hasse-Weil zeta function, which is closely related to the Hasse Zeta function of a proper regular model $\mathfrak{X}$ of $X$ (which is a scheme of finite type over $\mathbb{Z}$). The former only depends on $X$, the generic fiber of $\mathfrak{X}$, and the latter depends also on the special fibers, especially at the bad primes. For convenience the definitions are: \begin{equation} \zeta_X(s)=\frac{\prod_{m=0}^{\text{dim(X)}}L(H^{2m}(X),s)}{\prod_{m=0}^{\text{dim}(X)-1}L(H^{2m-1}(X),s)} \end{equation} \begin{equation} \zeta_{\mathfrak{X}}(s)=\prod_{x\in\mathfrak{X}_0}\frac{1}{1-|k(x)|^{-s}}, \end{equation} where $\mathfrak{X}_0$ denotes the closed points of $\mathfrak{X}$ and $|k(x)|$ denotes the residue field at $x$. Again, both are defined in some half plane. Of course, we expect meromorphic continuation of both zeta functions, and by considering their quotient, which is a finite product over bad primes, it is easy to see that their continuations are equaivalent.
Moreover, we expect a functional equation of the Hasse zeta function of $\mathfrak{X}$, the arithmetic scheme, with respect to $s\mapsto\text{dim}(\mathfrak{X})-s$. This should be equivalent to the functional equations of each Hasse-Weil $L$ or, indeed, of the Hasse-Weil zeta.
I have two questions, the first of which is how does one modify the conductor of $X$ to deduce an appropriate conductor for $\mathfrak{X}$? I know the process for curves and arithmetic surfaces.
The second question is, is it possible to prove that the quotient $\zeta_{\mathfrak{X}}(s)/\zeta_X(s)$ is invariant with respect to $s\mapsto\text{dim}(\mathfrak{X})-s$? Again, I understand the argument for curves.
