Fourier expansion of Arakawa lifting II: Relation with central L-values
This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317–369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic [Formula: see text]-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central [Formula: see text]-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central [Formula: see text]-value. We can relate the square norm with the central value of some [Formula: see text]-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the [Formula: see text]-functions in our concern.