First moments of GL(3)×GL(2) and GL(2)L‐functions and their applications

Abstract Let be a self‐dual Hecke–Maaß form for underlying the symmetric square lift of a ‐newform of square‐free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of ‐functions and ‐functions. As a result, we obtain an estimate for the first moment for in a family, where is of the level , and for any primes such that . We prove the subconvex bound for involving the level aspects simultaneously in the range and for any for the first time. Moreover, we further investigate the first moments of these ‐functions in the weight aspect over , with being a large number. As the results, we obtain a Lindelöf average bound for the first moment of of degree 8 and an asymptotic formula for the first moment of with an error term of , respectively.