An Inner Product Relation on Saito-kurokawa Lifts

Let f be a newform of weight 2k−2 and level M with M an odd square-free integer. Via the Saito-Kurokawa correspondence there is associated to f a Siegel newform Ff of weight k and level M . In this paper we provide a formula relating the Petersson products 〈Ff , Ff 〉 and 〈f, f〉. We use this result to give a new proof of a special case of a well-known result of Shimura on the algebraicity of a special value of a Rankin convolution L-function.