Determination of a pair of newforms from the product of their twisted central values

Abstract

We show that a pair of newforms (f,g)(f,g) can be uniquely determined by the product of the central 𝐿-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted 𝐿-functions, L(1/2,f×χ)L(1/2,g×χψ)L⁢(1/2,f×χ)⁢L⁢(1/2,g×χ⁢ψ) , where (f,g)(f,g) is a pair of newforms. The average is taken over the primitive Dirichlet characters 𝜒 and 𝜓 of distinct prime moduli.