Twists and Resonance ofL‐Functions, II

Abstract

Abstract. We continue our investigations of the analytic properties of nonlinear twists of L-functions developed in [4], [5] and [7]. Let F (s) be an L-function of degree d. First we extend the transformation formula in [5], relating a twist F (s; f ) with leading exponent κ 0 > 1/d to its dual twist F (s; f * ). Then we combine the results in [7] with such a transformation formula to obtain the analytic properties of new classes of nonlinear twists. This allows to detect several new cases of resonance of the classical L-functions.Mathematics Subject Classification (2000): 11 M 41

Classification of L-functions of degree 2 and conductor 1

“…Here θ F is the internal shift defined in Section 2 and f * is the dual of f . The precise shape of f * is important in this lemma, but only a general description of f * is given in [9]; see Section 1.3 there. So we first proceed with the explicit computation of f * .…”

“…The first step requires the computation of the z-critical point x 0 = x 0 (ξ) of the function Φ(z, ξ) defined by equation (1.5) of [9], which in our case becomes…”

Nonlinear twists and moments of $L$-functions

“…We shall briefly recall the basic notation and results in Section 2. In our papers [1,[4][5][6][7] we studied the analytic properties of a class of nonlinear twists of F . Moreover, in these and other papers, notably in [8,9], we refined and applied such properties to the study of the structure of the Selberg classes.…”