is not accessible via the Langlands-Shahidi method. Some information can be obtained through the relationship with the better-understood Rankin-Selberg $L$-function. In particular, one can get a global functional equation of the adjoint $L$-function $L(s,\pi,{{\mathrm{Ad}}}\times \chi)$ by the global functional equations of $L(s,\pi\times {\widetilde}\pi\times \chi)$ and $L(s,\chi)$. To obtain further information of the adjoint $L$-function in the case $n=3$, our main tool here is an integral representation pioneered in [@G] together with a method of ruling out poles pioneered in [@GJ].

Ginzburg’s local zeta integral is based on an embedding of ${{\mathrm{SL}}}_3$ into a split exceptional group of type $G_2.$ The argument was adapted in [@H12] to apply to special quasisplit unitary group ${{\mathrm{SU}}}_{2,1}$ (which depends on a fixed quadratic extension of $F$), which also embed into $G_2.$ Let $H$ be one of these groups, regarded as a subgroup of $G_2.$ Then Ginzburg’s global zeta integral is given by $$Z(\varphi, f_s)=\int_{H(F) {\backslash}H({\mathbb{A}})} \varphi(g) E(g,f_s) \, dg,$$ where $\varphi$ is a cuspidal automorphic form on ${{\mathrm{GL}}}_3({{\mathbb {A}}})$ or ${{\mathrm {U}}}_{2,1}({{\mathbb {A}}})$ in a fixed irreducible cuspidal automorphic representation $\pi$ and $E(g, f_s)$ is an Eisenstein series on $G_2$. See [@G; @H18] or $\S$\[sec: split main theorem\] for more details. In the split 