Nonvanishing of Central Derivatives of Modular $L$-series in Level $p^2$

Abstract

A quadratic twist of the L-function associated with a modular form is known to satisfy a functional equation, which may be even or odd. A result due to Gross and Zagier explicitly computes the central value of the L-function or its derivative. In prime level when the functional equation is even, Michel and Ramakrishnan have used an averaging method to prove several consequences of the Gross-Zagier formulae, including a non-vanishing result. The present research concerns L-functions arising from newforms in prime-squared level, which necessarily have odd functional equations. Such an L-function has a central value of zero; the Gross-Zagier formulae compute the central value of its derivative. Using the Michel-Ramakrishnan averaging method, we compute the average value of these derivatives over different L-functions. In particular, we show that under suitable conditions there exists an L-function with only a simple zero at the center of symmetry. The proof requires us to bound the contribution of oldforms from level $p$.