Petersson products of modular forms associated with the values of \(L\)-functions (Q1396438)

Let \(p\) be a prime, \(\chi \) a Dirichlet character \(\pmod{p}\), \(f(z)=\sum_{n=1}^{\infty }a_ne^{2\pi inz}\) a cusp form of weight 2 for the congruence group \(\Gamma _0(p)\), and \(L(f,\chi ,s) = \sum_{n=1}^{\infty }\chi (n)a_nn^{-s}\) the corresponding \(L\)-function twisted with the character \(\chi \). The mapping \(f\mapsto L(f,\chi ,1)\), which is a linear form in the space \(S_2(\Gamma _0(p))\), can be represented as the inner product against a certain unique form \(h_{\chi }\in S_2(\Gamma _0(p))\), thus \(L(f, \chi ,1)=(f, h_{\chi })\). The main result in this paper is an identity for the inner product \((h_{\chi }, h_{\chi '})\) if \(\chi \) and \(\chi '\) are non-trivial characters of different parity. This identity involves Gauss sums and values of Dirichlet \(L\)-functions at \(s=1\).