Let F/Fobe a quadratic extension of non-archimedean locally compact fields of odd residualcharacteristic andσbe its non-trivial automorphism. We show that anyσ-self-dual cuspidal re-presentation of GLn(F) contains aσ-self-dual Bushnell–Kutzko type. Using such a type, we cons-truct an explicit test vector for Flicker’s local Asai L-function of a GLn(Fo)-distinguished cus-pidal representation and compute the associated Asai root number. Finally, by using global me-thods, we compare this root number to Langlands–Shahidi’s local Asai root number, and moregenerally we compare the corresponding epsilon factors for any cuspidal representation.