Interpolating local constants in families

We extend the theory of local constants to l-adic families of representations of GL_n(F) where F is a p-adic field with l not equal to p. We construct zeta integrals and gamma factors for representations coming from the conjectural “local Langlands correspondence in families” of Emerton-Helm, proving a rationality result and functional equation. We also construct a universal gamma factor with coefficients in the integral Bernstein center.