On the existence of a p-adic metaplectic Tate-type-factor

Let F be a p-adic field, let χ be a character of F^*, let ψ be a character of F and let γ_ψ be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic function (χ, s, ψ) via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against ψ. γ_ψ^-1. It turns out that γ and have similar integral representations. Furthermore, has a relation to Shahidi's metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi's local coefficient. Up to an exponential factor, (χ, s, ψ) is equal to the ratio.