Petersson scalar products and L-functions arising from modular forms

Let f(z)=∑n=0∞ane(nz),g(z)=∑n=0∞bne(nz)(e(z)=e2π-1z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=\sum _{n=0}^{\infty }a_{n}{\mathbf {e}}(nz),g(z)=\sum _{n=0}^{\infty }b_{n}{\mathbf {e}}(nz)\ ({\mathbf {e}}(z)=e^{2\pi \sqrt{-1}z})$$\end{document} be holomorphic modular forms for Γ0(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{0}(N)$$\end{document} of integral weight or half integral weight, where their weights or characters are not necessarily equal to each other. We show that L(s;f,g):=∑n=1∞anb¯nn-s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s;f,g):=\sum _{n=1}^{\infty }a_{n}{\overline{b}}_{n}n^{-s}$$\end{document} extends meromorphically to the whole s plane, and that it satisfies some functional equation. Some residues of the L-function or some special values are expressed in terms of the Petersson scalar product. Applications to quadratic forms are included.