If $X$ is a scheme of finite type over $\mathbb{Z}$, and $X\_0$ denotes its set of closed points, then one can define its zeta function on the half plane $Re(s)>\text{dim}(X)$:
\begin{equation}
\zeta\_X(s)=\prod\_{x\in X\_0}\frac{1}{1-|k(x)|^{-s}},
\end{equation}
where $k(x)$ denotes the (finite) residue field at $x$.


The zeta function has a completion, which includes the conductor and a gamma factor. I want to ask about the latter, in a special case, namely when $X$ is a (proper, regular) model of an abelian variety over a number field.


For example, if $X$ is a model of an elliptic curve $E$ over a number field $k$, then the zeta function of $X$ can be written:
\begin{equation}
\zeta\_X(s)=n\_{X}(s)\frac{\zeta\_k(s)\zeta\_k(s-1)}{L\_E(s)},
\end{equation}
where $n\_X(s)$ depends only on the bad primes. Indeed, an identical expression holds for a model of any smooth projective curve over a number field. The quotient on the right hand side of the above expression is what some call the Hasse--Weil $\zeta$-function of $E$. This notion can be extended to any smooth projective variety $V$ of dimension $n$:
\begin{equation}
\frac{\prod\_{i=0}^nL(H^{2i}\_{et}(V),s)}{\prod\_{i=0}^{n-1}L(H^{2i+1}\_{et}(V),s)}.
\end{equation}
The gamma factor for the zeta function of a model of $V$ is the alternating product of the gamma factors for each Hasse--Weil L-factor (coming from the Hodge decomposition of each cohomology group). In the case of elliptic curves, direct computation shows that the gamma factor for a model is in fact a rational function. I have been lead to believe that this is true for any abelian variety, but I cannot prove it.


Firstly, is it true that the gamma factor of a model $\mathcal{A}\rightarrow\text{Spec}(\mathcal{O}\_k)$ of an abelian variety $A$ over a number field $k$ is rational? Secondly, is there a profound reason that this should be true? 



Gamma Factors for Zeta Functions of Abelian Varieties
This amounts to an Euler characteristic argument. A ratio of products of Gamma functions such as 
$$ \frac{\prod\_{i=1}^r \Gamma(s - a\_i)}{\prod\_{j=1}^t \Gamma(s - b\_j)} $$
is rational if and only if $r = t$ (i.e. there are as many functions in the numerator as in the denominator). But the number of $\Gamma$ factors in the numerator of the zeta function is the sum of the dimensions of the even-degree cohomology groups, and the number of factors in the denominator is the sum of the odd-degree ones. So the ratio is a rational function of $s$ if and only if the Euler characteristic is 0. 


However, all Abelian varieties have Euler characteristic 0 (since over $\mathbf{C}$ they are products of circles) and that does it.


