  Under a hypothesis which is slightly stronger than the Riemann Hypothesis for
elliptic curve $L$-functions, we show that both the average analytic rank and
the average algebraic rank of elliptic curves in families of quadratic twists
are exactly $\frac 12$. As a corollary we obtain that under this last
hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all
curves in our family, and that asymptotically one half of these curves have
algebraic rank $0$, and the remaining half $1$. We also prove an analogous
result in the family of all elliptic curves. A way to interpret our results is
to say that nonreal zeros of elliptic curve $L$-functions in a family have a
direct influence on the average rank in this family. Results of Katz-Sarnak and
of Young constitute a major ingredient in the proofs.
