Gamma factors for the Asai cube representation

We prove an equality between the gamma factors for the Asai cube representation of ${\rm R}_{E/F}{\rm GL}_2$ defined by the Weil$-$Deligne representations and the local zeta integrals of Ikeda and Piatetski-Shapiro$-$Rallis, where $E$ is an etale cubic algebra over a local field $F$ of characteristic zero. As an application we obtain the analytic properties of the automorphic $L$-functions for the Asai cube representation.