In [J. Inst. Math. Jussieu 14 (2015), 149–184] and [Int. Math. Res. Not. IMRN 7 (2017), 2014–2099] a family of Rankin-Selberg integrals was shown to represent the twisted standard
L
\mathcal {L}
-function
L
(
s
,
π
,
χ
,
s
t
)
\mathcal {L}(s,\pi ,\chi ,\mathfrak {st})
of a cuspidal representation
π
\pi
of the exceptional group of type
G
2
G_2
. These integral representations bind the analytic behavior of this
L
\mathcal {L}
-function with that of a family of degenerate Eisenstein series for quasi-split forms of
S
p
i
n
8
Spin_8
associated to an induction from a character on the Heisenberg parabolic subgroup.
This paper is divided into two parts. In Part 1 we study the poles of these degenerate Eisenstein series in the right half-plane
R
e
(
s
)
>
0
\mathfrak {Re}(s)>0
. In Part 2 we use the results of Part 1 to prove the conjecture, made by J. Hundley and D. Ginzburg in [Israel J. Math. 207 (2015), 835–879], for stable poles and also to give a criterion for
π
\pi
to be a CAP representation with respect to the Borel subgroup of
G
2
G_2
in terms of the analytic behavior of
L
(
s
,
π
,
χ
,
s
t
)
\mathcal {L}(s,\pi ,\chi ,\mathfrak {st})
at
s
=
3
2
s=\frac {3}{2}
.