Low-lying zeros of symmetric power 𝐿-functions weighted by symmetric square 𝐿-values

Abstract

Given a totally real number field 𝐹 and its adèle ring 𝔸_𝐹, let 𝜋 vary in the set of irreducible cuspidal automorphic representations of PGL₂(𝔸_𝐹) corresponding to primitive Hilbert modular forms of a fixed weight. We determine the symmetry type of the one-level density of low-lying zeros of the symmetric power 𝐿-functions 𝐿(𝑠, Sym^𝑟(𝜋)) weighted by special values of the symmetric square 𝐿-functions 𝐿((𝑧+1)/2, Sym²(𝜋)) at 𝑧 ∈ [0, 1] in the level aspect. If 0 < 𝑧 ≤ 1, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power 𝐿-functions for 𝐹 = ℚ with harmonic weight. Hence our result is regarded as a 𝑧-interpolation of Ricotta and Royer's result. If 𝑧 = 0, the density of low-lying zeros weighted by central values is of a different type only when 𝑟 = 2.