  It is believed that, in the limit as the conductor tends to infinity,
correlations between the zeros of elliptic curve $L$-functions averaged within
families follow the distribution laws of the eigenvalues of random matrices
drawn from the orthogonal group. For test functions with restricted support,
this is known to be the true for the one- and two-level densities of zeros
within the families studied to date. However, for finite conductor Miller's
experimental data reveal an interesting discrepancy from these limiting
results. Here we use the L-functions ratios conjectures to calculate the
1-level density for the family of even quadratic twists of an elliptic curve
L-function for large but finite conductor. This gives a formula for the leading
and lower order terms up to an error term that is conjectured to be
significantly smaller. The lower order terms explain many of the features of
the zero statistics for relatively small conductor and model the very slow
convergence to the infinite conductor limit.
  However, our main observation is that they do not capture the behaviour of
zeros in the important region very close to the critical point and so do not
explain Miller's discrepancy. This therefore implies that a more accurate model
for statistics near to this point needs to be developed.
