  In this paper, we obtain an unconditional density theorem concerning the
low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From
this together with the Riemann hypothesis for these L-functions, we infer the
majorant of 27/14 (which is strictly less than 2) for the average rank of the
elliptic curves in the family under consideration. This upper bound for the
average rank enables us to deduce that, under the same assumption, a positive
proportion of elliptic curves have algebraic ranks equaling their analytic
ranks and finite Tate-Shafarevic group. Statements of this flavor were known
previously under the additional assumptions of GRH for Dirichlet L-functions
and symmetric square L-functions which are removed in the present paper.
