Conditional lower bounds on the distribution of central values: the case of modular forms

Radziwill and Soundararajan unveiled a connection between low-lying zeros and central values of $L$-functions, which they instantiated in the case of quadratic twists of an elliptic curve. This paper addresses the case of the family of modular forms in the level aspect, and proves that the logarithms of central values of associated L-functions approximately distribute along a normal law with mean -(1/2)log log c(f) and variance log log c(f), where c(f) is the analytic conductor of f, as predicted by the Keating-Snaith conjecture.