Let $X$ be a smooth projective toric variety over a number field $K$ (assume the tori is split). As $X$ is rational, maybe the related HasseWeil zeta function can be wellunderstand, so how much do we know about the zeta function $L(X,s)$ from the fan (at least for some twist forms of $X$)? How about its special values?
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$\begingroup$ There are results on the motive associated to a toric variety, see Theorem 3.5 in arxiv.org/abs/math/0407305 See also Example 7 of math.mit.edu/~guozhen/motiv.pdf for a more precise result. $\endgroup$ – François Brunault Apr 6 at 22:44

3$\begingroup$ The cycle map from the Chow group of $X$ to $\ell$adic cohomology would be surjective (I can dig up a reference later). Therefore $H^{2i}(X,\mathbb{Q}_\ell)= Q(i)^{b_{2i}}$, for some $b_{2i}$, and the odd groups are zero. Therefore the Weil zeta function of the closed fibres is the product $\prod 1/(1q^it)^{b_{2i}}$. This is should be enough to assemble the HasseWeil, I think. $\endgroup$ – Donu Arapura Apr 6 at 22:45