is holomorphic and nonvanishing at $s=1,$ while $L^S(s, \pi, {{\mathrm{Ad}}}\times \chi_{E/F})$ can have at most a double pole. More precisely, we have three sets of equivalent conditions.

1.  The following are equivalent:

    1.  $L^S(s, \pi, {{\mathrm{Ad}}}\times \chi_{E/F})$ is holomorphic and nonvanishing at $s=1$

    2.  the stable base change lift, ${{\operatorname}{sbc}}(\pi),$ of $\pi$ is cuspidal

    3.  $\pi$ is not endoscopic

    4.  $\pi$ has a nonzero ${{\mathrm{SU}}}_{1,1}$ period

2.  The following are equivalent:

    1.  $L^S(s, \pi, {{\mathrm{Ad}}}\times \chi_{E/F})$ has a simple pole at $s=1$

    2.  ${{\operatorname}{sbc}}(\pi)$ is the isobaric sum of a character and a cuspidal representation of ${{\mathrm{GL}}}_2({\mathbb{A}}_E)$

    3.  $\pi$ is an endoscopic lift from ${{\mathrm {U}}}_{1,1}({\mathbb{A}}) \times {{\mathrm {U}}}_1({\mathbb{A}}),$ but not from ${{\mathrm {U}}}_{1}({\mathbb{A}}) \times {{\mathrm {U}}}_{1}({\mathbb{A}}) \times {{\mathrm {U}}}_{1}({\mathbb{A}}).$

3.  The following are equivalent:

    1.  $L^S(s, \pi, {{\mathrm{Ad}}}\times \chi_{E/F})$ has a double pole at $s=1$

    2.  ${{\operatorname}{sbc}}(\pi)$ is the isobaric sum of three characters of ${\mathbb{A}}_E^\times,$

    3.  $\pi$ is an endoscopic lift from ${{\mathrm {U}}}_{1}({\mathbb{A}}) \times {{\mathrm {U}}}_{1}({\mathbb{A}}) \times {{\mathrm {U}}}_{1}({\mathbb{A}}).$

The twist by $\chi_{E/F}$ is of special importance for at least two reasons. First, in the nonsplit case the local zeta integral $Z^*(W_v, f_{v,s})$ is equal to $L(s, \pi_v, {{\mathrm{Ad}}}\times \chi_{E/F,v}\chi_v)$ as opposed to $L(s, \pi_v, {{\mathrm{Ad}}}\times \chi_v).$ 