Certain Dirichlet series attached to automorphic forms over imaginary quadratic fields

Values of Twisted Tensor L-functions of Automorphic Forms Over Imaginary Quadratic Fields

“…We have f (γg) = f (g) for γ ∈ GL 2 (K), f (zg) = ψ n (z) f (g) for z ∈ Z A , and for γ 0 γ ∞ ∈ Γ 0 (n)SU 2 (C) we have f (gγ 0 γ ∞ )(S, T) = ψ n (γ 0 ) f (g)(γ ∞ (S, T)). Let f be an eigenfunction of the operators D σ as in [10] or [20]. In particular, D σ f = (n 2 σ /2 + n σ ) f as in [11].…”

“…Note that c f ( • ) can be considered to be a function on the fractional ideals of K that vanishes outside of the integral ideals. In the sequel we consider f to be a newform, a Hecke eigenfunction in the sense of [20,Section 4], and normalized so that c f (O K ) = 1. Let A be a Q(ψ n )-algebra and let L(n, A) be the space of homogeneous polynomials of degree n in x = (X, Y ) and degree n in…”

“…The L-function L T (s, f ) has a meromorphic continuation and functional equation from [20]. All critical values of L T (s, f ) in the right half of the critical strip were treated in [7].…”

“…Although the archimedean factors were not completely determined, the integral was explicit enough to determine some arithmetic properties of L T (s, f ) in the following way. For m ⊂ O K let T(m) be a Hecke operator as in [20,Section 4] and then T(m) f = λ f (T(m)) f , where λ f is the Hecke algebra character corresponding to f . Let E = Q( f , ψ n ).…”

“…A precise definition of L T (s, f ) is given in Section 3. Analytic properties of L T (s, f ) for certain cuspforms were studied by Takase [19] and Zhao [20]. Arithmetic properties of the L-functions studied here were investigated by Ghate in [6,7].…”