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

Abstract We show that a pair of newforms <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>f</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (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, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:mi>χ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mi>g</m:mi> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:mi>χ</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mi>ψ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L(1/2,f\times\chi)L(1/2,g\times\chi\psi) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>f</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (f,g) is a pair of newforms. The average is taken over the primitive Dirichlet characters 𝜒 and 𝜓 of distinct prime moduli.