Reference request: normalization of intertwining operators for GL(2, C)

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?

• Did you consult Arthur's early papers on the trace formula?
– user1688
Jan 8 '16 at 16:59
• 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. Jan 9 '16 at 14:37
• 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 non-vanishing holomorphic operator one normalizes by $\epsilon(0,\chi_1\chi_2^{-1})\frac{L(1,\chi_2\chi_1^{-1})}{L(0,\chi_1\chi_2^{-1})}.$ Dec 6 '16 at 17:30