Hello,


this may be a very naive question, but has the degree conjecture (namely "the degree of any function of the Selberg class is a non negative integer") been proven for automorphic L-functions?
Thank you in advance.



Degree conjecture and automorphic L-functions
Going off [wikipedia](http://en.wikipedia.org/wiki/Selberg_class#Basic_properties), it is true that automorphic $L$-functions for $GL\_n$ over a number field have non-negative integral degree, where by degree I mean the number $2\sum\_{i=1}^k \omega\_i$, where the $\omega\_i$ are the coefficients of $s$ appearing in the gamma factor, which is more-or-less
$$L\_\infty(s,F)=Q^s\prod\_{i=1}^k\Gamma(\omega\_i s+\mu\_i)$$
We know that for $GL\_n(\mathbb R)$ and $GL\_n(\mathbb C)$ the $\omega\_i$ are either $1/2$ or $1$ (see, e.g., Knapp's "Local Langlands Correspondence: The Archimedean case", in Motives, vol 2), so twice the sum will always be an integer (of course, only a few of these $L$-functions are known to be in Selberg's class).


For general $G$, it depends on whether someone has written done the $L$-factors for general real reductive groups. I don't know if this has been done, or if it is technically known but difficult to write out, or if it is not known.


