  Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution
considerations we define, for each prime $\ell$, a local factor which measures
the relative frequency with which $f\bmod \ell$ occurs as the characteristic
polynomial of a symplectic similitude over $\mathbb{F}_\ell$. For a certain
class of polynomials, we show that the resulting infinite product calculates
the number of principally polarized abelian surfaces over $\mathbb{F}_q$ with
Weil polynomial $f$.
