Tate γ -factor, Weil index, and the metaplectic γ~ -factor

Let F be a p-adic field, let ψ be a non-trivial character of F, and let χ be a character of F^∗. In this short note we present two new identities involving γ(s, χ, ψ) , γ(ψ) , and γ~ (s, χ, ψ) along with a duplication formula for γ(s, χ, ψ). Here γ(s, χ, ψ) is the Tate γ-factor, γ(ψ) is the Weil index, and γ~ (s, χ, ψ) is the metaplectic γ~ -factor. As a result we give a new proof for a useful identity involving these three factors originally proven by W. Jay Sweet.