Linear dependence of certain L ‐values of half‐integral weight modular forms

Let h be a cusp form of half‐integral weight, andε 2 / 3 ( χ )be a certain (not necessarily holomorphic) modular form of weight3 2associated with a Dirichlet character χ. We then prove linear dependence of the valuesR ( l , h , ε 2 / 3 ( χ ) )of the Rankin–Selberg convolution products of h andε 2 / 3 ( χ )when we fix an integer l and vary χ. A main idea for the proof is to express such a convolution product as the twisted Koecher–Maass series L ( s, M ( h ), χ) for the Maass lift M ( h ) of h , whose values at integers were investigated by Choie and Kohnen. Moreover, by using such an expression, we obtain an algebraicity result for the values of L ( s, M ( h ), χ) at half‐integers, which was not considered by Choie and Kohnen.