(See [@H12].) Second, the analogue of in the nonsplit case relates $L(s, \pi , {{\mathrm{Ad}}}\times \chi)$ to the Asai $L$-function $L(s, {{\operatorname}{sbc}}(\pi), {{\operatorname}{Asai}})$ of the stable base change lift ${{\operatorname}{sbc}}(\pi)$ ([@KimKrishOdd; @KimKrishEven]) of $\pi.$ Recall that ${{\operatorname}{sbc}}(\pi)$ is an automorphic representation of ${{\mathrm{GL}}}_3({\mathbb{A}}_E).$ The twist by $\chi_{E/F}$ is of particular importance because for a cuspidal representation $\Pi$ of ${{\mathrm{GL}}}_n({\mathbb{A}}_E),$ the $L(s, \Pi, {{\operatorname}{Asai}}) L(s, \Pi, {{\operatorname}{Asai}}\times \chi_{E/F})$ is equal to the Rankin-Selberg convolution $L$-function of $\Pi$ with the automorphic representation of ${{\mathrm{GL}}}_3({\mathbb{A}}_E)$ obtained by composing $\Pi$ with the nontrivial element of ${{\mathrm{Gal}}}(E/F).$ (cf [@GRSBook p.317]). We also obtain results which characterize when an adjoint $L$ function twisted by a character other than $\chi_{E/F}$ can have a pole. See theorems \[thm:other poles; type3\], \[thm:other poles: type 2+1\] and proposition \[prop: other poles, type 1+1+1\].

Because it relies on an embedding of ${{\mathrm{SL}}}_3$ into $G_2$, Ginzburg’s integral representation does not generalize to ${{\mathrm{GL}}}_n$ for general $n.$ However, it was shown in [@BumpGinzburg] that a certain global integral in the split exceptional group $F_4$ represents the adjoint $L$-function of ${{\mathrm{GL}}}_4,$ and some evidence was presented in [@GH3] that a certain global integral in the exceptional group $E_8$ represents the adjoint $L$-function for ${{\mathrm{GL}}}_5.$ 