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? 



Reference request: normalization of intertwining operators for GL(2, C)
This is on page 57-58 of M. Duflo, "Representations irreductibles des groupes semisimples complexes", in "Analyse harmonique sure les groupes de Lie", SLN 497, 1975, 26-88. 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, 1343-1394, 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> )


