A p-adic analog of Hasse - Davenport product relation involving ∈-factors

In this paper we prove some generalizations of the classical Hasse-Davenport product relation for certain arithmetic factors defined on a p-adic field F, among them one finds the ∈-factors appearing in Tate's thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL₂(F) to Plancherel measures and γ-factors.