The second moment of $GL(4) \times GL(2)$ $L$-functions at special points

In this paper, we obtain upper bounds for the second moment of L(uj × φ, 1 2 + itj), where φ is a Hecke Maass form for SL(4,Z), and uj is taken from an orthonormal basis of Hecke-Maass forms on SL(2,Z) with eigenvalue 1/4 + t2j . The bounds are consistent with the Lindelöf hypothesis. Previously these types of upper bounds are available for only GL(n)×GL(2), where n ≤ 3.

[1]  Xiannan Li.  Upper bounds on L-functions at the edge of the critical strip , 2009, 0904.0850.

[2]  Xiannan Li,et al.  The second moment of GL(4)×GL(2) L-functions at special points , 2020 .

[3]  W. Luo.  The spectral mean value for linear forms in twisted coefficients of cusp forms , 1995 .

[4]  Henry H. Kim,et al.  FUNCTORIALITY FOR THE EXTERIOR SQUARE OF GL4 AND THE SYMMETRIC FOURTH OF GL2 , 2003 .

[5]  Vorrapan Chandee,et al.  THE EIGHTH MOMENT OF THE FAMILY OF Γ1(q)-AUTOMORPHIC L-FUNCTIONS , 2018 .

[6]  H. Iwaniec,et al.  Analytic Number Theory , 2004 .

[7]  H. Iwaniec,et al.  The orthogonality of Hecke eigenvalues , 2007, Compositio Mathematica.

[8]  W. Schmid,et al.  A general Voronoi summation formula for GL(n,Z) , 2009, 0912.1065.

[9]  Xiannan Li,et al.  The sixth moment of automorphic $L$-functions , 2017, 1708.08406.

[10]  Fritz Oberhettinger,et al.  Tables Of Bessel Transforms , 1961 .

[11]  Xiaoqing Li.  The Central value of the Rankin–Selberg L-Functions , 2008, 0812.0035.

[12]  R. Phillips,et al.  On cusp forms for co-finite subgroups ofPSL(2,ℝ) , 1985 .

[13]  A. Ivic.  Sieve Methods, Exponential Sums, and their Applications in Number Theory: On the Ternary Additive Divisor Problem and the Sixth Moment of the Zeta-Function , 1997 .

[14]  P. Sarnak,et al.  On Selberg's eigenvalue conjecture , 1995 .

[15]  Xiannan Li,et al.  THE EIGHTH MOMENT OF DIRICHLET L-FUNCTIONS , 2013, 1303.4482.

[16]  M. Abramowitz,et al.  Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables , 1966 .

[17]  L. Debnath.  Tables of Integral Transforms , 2012 .

[18]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.