by conjugation, we obtain an action on ${\mathfrak}{gl}_3({\mathbb{C}}).$ The subspace ${\mathfrak}{sl}_3({\mathbb{C}})$ is preserved. We denote the action of $^LH$ on ${\mathfrak}{sl}_3({\mathbb{C}})$ by ${{\mathrm{Ad}}}.$

In this paper we study the twisted adjoint $L$-function $L(s, \pi, {{\mathrm{Ad}}}\times \chi)$ where $\chi$ is a Hecke character. This work is motivated by the following simple conjecture regarding the untwisted $L$-function in the split case.

\[conj: adjoint GL3 is entire\] Let $\pi$ be an irreducible cuspidal automorphic representation of ${{\mathrm{GL}}}_n({\mathbb{A}})$, where ${\mathbb{A}}$ is the adele ring of a global field. Let ${{\mathrm{Ad}}}$ denote the adjoint representation of ${{\mathrm{GL}}}_n({{\mathbb {C}}})$ on ${\mathfrak}{sl}_n({\mathbb{C}}).$ Then the global Langlands $L$-function $L(s, \pi, {{\mathrm{Ad}}})$ is entire.

In the case $n=2,$ Conjecture \[conj: adjoint GL3 is entire\] was proved by Gelbart-Jacquet [@GelbartJacquet], generalizing a method of Shimura [@Shimura]. A different proof was given in [@JZ]. Flicker [@Flicker] proved that under a certain assumption about ratios of Hecke $L$ series (see “$({\operatorname}{Ass}; E, \omega),$” [@Flicker p. 233]), Conjecture \[conj: adjoint GL3 is entire\] holds if $\pi\cong \otimes_v \pi_v$ with at least one $\pi_v$ supercuspidal. More precisely, he proved that for such $\pi,$ the twisted adjoint $L$-function $L(s, \pi, {{\mathrm{Ad}}}\times \chi)$ is entire unless $\chi$ is nontrivial and $\pi \cong \pi \otimes \chi.$ This 