Let f be an Hecke eigenform for the group Γ 0 (q) and χ d be a primitive quadratic character of conductor |d|. In this article, we prove an asymptotic for the second moment of the derivative of L(s, f ⊗ χ 8d ) at the central point 1/2, which was previously known under GRH by Petrow [8].

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In this article, we will prove subconvex bounds for GL(3) × GL(2) L-functions in depth aspect.

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx. 22 (2010) 403–419].