On the characteristic polynomial of the Frobenius on \'etale cohomology

Let $X$ be a smooth proper variety of even dimension $d$ over a finite field. We establish a restriction on the value at $(-1)$ of the characteristic polynomial of the Frobenius on the middle-dimensional 'etale cohomology of $X$ with coefficients in ${\mathbb{Q}}_l(d\mathrm{/}2)$.