  By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in
${\mathbb P}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic
such hypersurface has a reciprocal root distinguished by minimal
$p$-divisibility. We study the $p$-adic variation of that distinguished root in
a family and show that it equals the product of an appropriate power of $p$
times a product of special values of a certain $p$-adic analytic function
${\mathcal F}$. That function ${\mathcal F}$ is the $p$-adic analytic
continuation of the ratio $F(\Lambda)/F(\Lambda^p)$, where $F(\Lambda)$ is a
solution of the $A$-hypergeometric system of differential equations
corresponding to the Picard-Fuchs equation of the family.
