  Using the ratios conjectures as introduced by Conrey, Farmer and Zirnbauer,
we obtain closed formulas for the one-level density for two families of
L-functions attached to elliptic curves. From those closed formulas, we can
determine the underlying symmetry types of the families. The one-level scaling
density for the first family corresponds to the orthogonal distribution as
predicted by the conjectures of Katz and Sarnak, and the one-level scaling
density for the second family is the sum of the Dirac distribution and the even
orthogonal distribution. This seems to be a new phenomenon, caused by the fact
that the curves of the second families have odd rank. Then, there is a trivial
zero at the central point which accounts for the Dirac distribution, and also
affects the remaining part of the scaling density which is then (maybe
surprisingly) the even orthogonal distribution. The one-level density for this
second family was studied in the past for test functions with Fourier
transforms of small support, but since the Fourier transforms of the even
orthogonal and odd orthogonal distributions are undistinguishable for small
support, it was not possible to identify the distribution with those
techniques. This can be done with the ratios conjectures, and it sheds more
light on "independent" and "non-independent" zeroes and the repulsion
phenomenon.
