χ (mod q) |L(χ, 1/2)| 4 = P (log q) + O q − 5 512 +ε , for any ε > 0 and where the symbol * means that we avoid χ = 1, φ * (q) = q−2 is the number of primitive characters modulo q, P is a degree four polynomial with leading coefficient (2π 2 ) −1 and −5/512 = (1 − 2θ)1/80 with θ = 7/64 is the best known approximation towards the Ramanujan-Petersson conjecture and it is due to Kim and Sarnak [K + 03]. More recently, Blomer, Fouvry, Kowalski, Michel and Milićević revisited the problem in [BFK + 14] by considering more general moments, namely of the form M f,g (q) := 1 φ * (q) * χ (mod q)

Let F be a number field with adele ring AF, q, l two coprime integral ideals with q squarefree and π1, π2 two fixed unitary automorphic representations of PGL2(AF) unramified at all finite places. In this paper, we use regularized integrals to obtain a formula that links the first moment of L(π ⊗ π1 ⊗ π2, 1 2 ) twisted by the Hecke eigenvalues λπ(l), where π runs through unitary automorphic representations of PGL2(AF) with conductor dividing q, with some spectral expansion of periods over representations of conductor dividing l. In the special case where π1 = π2 = σ, this formula becomes a reciprocity relation between moments of L-functions. As applications, we obtain a subconvex estimate in the level aspect for the central value of the triple product L(π ⊗ π1 ⊗ π2, 1 2 ) and a simultaneous non-vanishing result for L(Sym 2 (σ) ⊗ π, 1

We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1 2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair (ω1, ω2) of multiplicative characters modulo q, there is a positive proportion of χ (mod q) such that L(χ, 1 2 ), L(χω1, 1 2 ) and L(χω2, 1 2 ) are simultaneously not too small. ContentsTheorem 1.3. -Let q > 2 be a prime number, ω 1 , ω 2 be Dirichlet characters of modulus q, f a primitive Hecke cusp form of level 1 or q and trivial nebentypus. Assume that f satisfies the Ramanujan-Petersson conjecture, then for any ε > 0, we havewhere the implied constant only depends on ε > 0 and polynomially on the Archimedean parameters of f (the weight or the Laplace eigenvalue) in (1.4).Remark 1.4. -The case where f is of level one and ℓ = 1 has been announced by S. Das and S. Ganguly and it seems that their method is similar to our. When f is of level q with non trivial central character, the proof of Theorem 1.3 is similar but requires a mild extension of [KMS15, Theorem 1.3] for Kloosterman sums twisted by characters. We will return to this question in a coming paper. Remark 1.5. -The asymptotic formula (1.4) is similar to the mixted cubic moment evaluated by S. Das and R. Khan in [DK14]

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