The functional equation of an Artin L-function has an elementary function ε(ρ,s) appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε(ρ,s) can be written in a canonical way as a product of local constants ε(ρv, s, ψv) associated to primes v. Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis.
Dwork (1956) proved the existence of the local constant ε(ρv, s, ψv) up to sign.
Deligne (1973) later discovered a simpler proof using global methods.
The local constants ε(ρ, s, ψE) depend on a representation ρ of the Weil group and a choice of character ψE of the additive group of E. They satisfy the following conditions: Brauer's theorem on induced characters implies that these three properties characterize the local constants.
Deligne (1976) showed that the local constants are trivial for real (orthogonal) representations of the Weil group.