Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ of dimension $n$. The Weil conjectures assert that the zeta function $Z(X_0,t)$ satisfies the functional equation

$$Z(X_0,t) = \pm q^{\frac{-n\chi(X)}{2}} t^{-\chi(X)} Z(X_0,q^{-n} t^{-n})$$

where $\chi(X) = \sum_i (-1)^i \dim H^i(X,\mathbf{Q}_\ell)$.

For the Riemann zeta function, there is a constant $c = \Gamma(s/2)\pi^{-s/2}$ which can be understood as a contribution by the local factor in the Euler product coming from the Archimedean place at $\infty$. Once this is incorporated, we get the nicer formula $\xi(s) = \xi(1 - s)$.

Is there a similar interpretation for the constant $c = \pm q^{\frac{-n\chi(X)}{2}} t^{-\chi(X)}$ in terms of a local factor in the Euler product? It seems that there is no missing place in the function field setting, however.