On Motohashi’s formula

We complement and offer a new perspective of the proof Motohashi-type formula relating fourth moment L L -functions for G upper L 1"> G 1 encoding="application/x-tex">GL_1 with third 2"> 2 encoding="application/x-tex">GL_2 over number fields, studied earlier by Michel-Venkatesh Nelson. Our main tool is type pre-trace test functions on M 2 left-parenthesis double-struck A right-parenthesis"> M ( A stretchy="false">) encoding="application/x-tex">M_2(\mathbb {A}) instead encoding="application/x-tex">GL_2(\mathbb , whose spectral side matrix coefficients are replaced standard Godement-Jacquet zeta integrals. This also generalization Bruggeman-Motohashi’s other Motohashi’s formula. give variation our method in case division quaternion algebras encoding="application/x-tex">M_2 yielding reciprocity, which we not sure if it within period formalism given Michel-Venkatesh. indicate further possible generalization, seems to be beyond what can offer.