  Recently Conrey, Farmer and Zirnbauer conjectured formulas for the averages
over a family of ratios of products of shifted L-functions. Their L-functions
Ratios Conjecture predicts both the main and lower order terms for many
problems, ranging from n-level correlations and densities to mollifiers and
moments to vanishing at the central point. There are now many results showing
agreement between the main terms of number theory and random matrix theory;
however, there are very few families where the lower order terms are known.
These terms often depend on subtle arithmetic properties of the family, and
provide a way to break the universality of behavior. The L-functions Ratios
Conjecture provides a powerful and tractable way to predict these terms. We
test a specific case here, that of the 1-level density for the symplectic
family of quadratic Dirichlet characters arising from even fundamental
discriminants d \le X. For test functions supported in (-1/3, 1/3) we calculate
all the lower order terms up to size O(X^{-1/2+epsilon}) and observe perfect
agreement with the conjecture (for test functions supported in (-1, 1) we show
agreement up to errors of size O(X^{-epsilon}) for any epsilon). Thus for this
family and suitably restricted test functions, we completely verify the Ratios
Conjecture's prediction for the 1-level density.
