  Fix an elliptic curve $E/\Q$, and assume the generalized Riemann hypothesis
for the $L$-function $ L(E_D, s) $ for every quadratic twist $E_D$ of $E$ by
$D\in\Z$. We combine Weil's explicit formula with techniques of Heath-Brown to
derive an asymptotic upper bound for the weighted moments of the analytic rank
of $E_D$. It follows from this that, for any unbounded increasing function $f$
on $\R$, the analytic rank and (assuming in addition the Birch-Swinnerton-Dyer
conjecture) the number of integral points of $E_D$ are less than $f(D)$ for
almost all $D$. We also derive an upper bound for the density of low-lying
zeros of $L(E_D, s)$ which is compatible with the random matrix models of Katz
and Sarnak.
