Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral $$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&1\\ 1& 0 \end{pmatrix} \begin{pmatrix} 1& x \\ 0 & 1 \end{pmatrix}g \right) \, dx.$$ (Which varies meromorphically as a function of $\chi_1,$ and $\chi_2$.) I would expect that if $L(\eta)$ denotes the usual local $L$ function attached to a character $\eta,$ then $L(\frac{\chi_1}{\chi_2})^{1} M(\chi_1, \chi_2)$ is holomorphic and never the zero operator. That is, if I've formulated correctly this is a pretty easy check when $F$ is nonarchimedean and immediate from the explicit formula given in (6.15) of Bump's Automorphic forms and representations if $F=\mathbb R.$ So, I imagine it holds and is well known for $\mathbb C$ as well, but does anyone have a reference?
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$\begingroup$ Did you consult Arthur's early papers on the trace formula? $\endgroup$– user1688Jan 8, 2016 at 16:59

$\begingroup$ A little. I didn't come across anything GL(2) (or SL(2)) specific. It seemed like this simplest case would probably be simpler and older. $\endgroup$– Joseph HundleyJan 9, 2016 at 14:37

$\begingroup$ Why do you expect the normalized operator, as you have defined, to be nonzero? Shouldn't it have a zero at the special representations, i.e. $\chi_1\chi_2^{1}=p^{\pm1}$? Most probably to get a nonvanishing holomorphic operator one normalizes by $\epsilon(0,\chi_1\chi_2^{1})\frac{L(1,\chi_2\chi_1^{1})}{L(0,\chi_1\chi_2^{1})}.$ $\endgroup$– Subhajit JanaDec 6, 2016 at 17:30
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1 Answer
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This is on page 5758 of M. Duflo, "Representations irreductibles des groupes semisimples complexes", in "Analyse harmonique sure les groupes de Lie", SLN 497, 1975, 2688. V. Drinfeld also noted in email to me (2014) that Zelobenko wrote up SL(2,C) in 1963, in Izv. Akad. Nauk SSSR Ser. Mat. 27, 13431394, but I did not corroborate the latter. Wallach states the outcome for SL(2,C) in his article in Corvallis.
(A more recent working of this exercise is at http://www.math.umn.edu/~garrett/m/v/intertwinings_SL2C.pdf )