Twists and resonance of $L$-functions, I

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. (2000): 11 M 41 Mathematics Subject Classification

Twists and Resonance ofL‐Functions, II

“…We start with an extension of the transformation formula, given in Theorem 1.1 of [5], relating a nonlinear twist F (s; f ) with κ 0 > 1/d to its dual twist F (s; f * ); see Theorem 1 below. Then we combine the results in [7] and Theorem 1 to obtain the analytic properties of new classes of nonlinear twists. Finally, we consider in greater detail some classes of twists of degree 2 L-functions.…”

“…This paper is a continuation of [7], to which we refer for definitions, notation and a general discussion of twists of L-functions. In [7] we gave a rather complete description of the analytic properties of the nonlinear twists (as usual e(x) = e 2πix ) In this paper we deal with the case κ 0 > 1/d.…”

“…This paper is a continuation of [7], to which we refer for definitions, notation and a general discussion of twists of L-functions. In [7] we gave a rather complete description of the analytic properties of the nonlinear twists (as usual e(x) = e 2πix ) In this paper we deal with the case κ 0 > 1/d. We start with an extension of the transformation formula, given in Theorem 1.1 of [5], relating a nonlinear twist F (s; f ) with κ 0 > 1/d to its dual twist F (s; f * ); see Theorem 1 below.…”

“…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 * ).…”

“…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.…”