Sums of Twisted Gl ( 3 ) Automorphic L - Functions

Let F be a number field and π be an automorphic representation on GLr (AF ). In this paper we consider weighted sums of quadratic twists of the L-function for π , ∑ d L(s, π, χd ) a(s, π, d) Nd −w , where χd is a quadratic character roughly attached to F( √ d)/F . We analyze the properties which the weights a(s, π, d) must satisfy if this is to satisfy a (certain, non-abelian) group of functional equations in (s, w), and show that there is a unique family of weight functions with this property for r ≤ 3. We describe these weights in detail when r = 3. As an application we give, for cuspidal π on GL3(AQ), a new proof of the holomorphicity of the symmetric square L-function of π . We also prove that if π ′ is a cuspidal automorphic representation of GL(2) over Q then infinitely many quadratic twists of the adjoint square L-function of π ′ are nonvanishing at the center of the critical strip. 0. Introduction. This paper concerns double Dirichlet series which may be expressed as weighted sums of quadratic twists of L-functions. In [BFH], a class of double Dirichlet series was proposed as follows. Let π be an automorphic representation of GL(r, A), where A is the adele ring of the global field F , let χπ be the central character of π , and let ω be an idèle class character of A×/F×. Roughly speaking, and restricting ourselves to the quadratic case, the Dirichlet series of interest is: Z0(s, w) = ∑ d L(s, π, χd) ω((d)) Nd −w (0.1) where d runs through classes of F× modulo squares, Nd is the absolute norm, χd is the quadratic character attached to F( √ d), and L(s, π, χd) denotes the twisted L-function of π . As is explained in [BFH], the expectation is that this function of two variables will satisfy two functional equations generating a finite group if r ≤ 3 and an infinite group if r ≥ 4. Unfortunately (0.1) is only an approximation to the actual Dirichlet series which we want. To explain why, let us sketch a method of studying such a series. First, each L-series being summed has a functional equation. Taking into account the power of the conductor of χd which occurs in the epsilon-factor for π ⊗ χd , one sees that there should be a functional equation for Z0(s, w) under the 1991 Mathematics Subject Classification. Primary 11F66, Secondary 11F70, 11M41, 11N75.