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It is believed that, in the limit as the conductor tends to infinity,correlations between the zeros of elliptic curve $L$-functions averaged withinfamilies follow the distribution laws of the eigenvalues of random matricesdrawn 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 zeroswithin the families studied to date. However, for finite conductor Miller'sexperimental data reveal an interesting discrepancy from these limitingresults. Here we use the L-functions ratios conjectures to calculate the1-level density for the family of even quadratic twists of an elliptic curveL-function for large but finite conductor. This gives a formula for the leadingand lower order terms up to an error term that is conjectured to besignificantly smaller. The lower order terms explain many of the features ofthe zero statistics for relatively small conductor and model the very slowconvergence to the infinite conductor limit.  However, our main observation is that they do not capture the behaviour ofzeros in the important region very close to the critical point and so do notexplain Miller's discrepancy. This therefore implies that a more accurate modelfor statistics near to this point needs to be developed.

Lower Order Terms for the One-Level Density of Elliptic Curve L-Functions