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 Lfunctions? Thank you in advance.
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 Lfunctions? Thank you in advance. 


Going off wikipedia, it is true that automorphic $L$functions for $GL_n$ over a number field have nonnegative 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 moreorless $$L_\infty(s,F)=Q^s\prod_{i=1}^k\Gamma(\omega_i s+\mu_i)$$ 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. 


To the best of my knowledge, we are very far from proving such a conjecture. If $\mathcal S_d$ is the subclass of the Selberg class $\mathcal S$ consisting of the functions of degree $d$ then it is known that 1) $\mathcal S_0=\{1\}$ (ConreyGhosh, 1993); 2) $\mathcal S_d=\emptyset$ for $0\lt d\lt1$ (Richert, 1957 and others); 3) $\mathcal S_1$ consists of the Riemann zeta function $\zeta(s)$ and the shifted Dirichlet $L$functions $L(s+i\tau,\chi)$ with $\tau\in\mathbb R$ and $\chi$ a primitive character (KaczorowskiPerelli, 1999); 4) $\mathcal S_d=\emptyset$ for $1\lt d\lt2$ (KaczorowskiPerelli, 2002 and 2011). Apart from these results, I think that nothing has been established in general. A nice survey of the results obtained so far can be found in the introduction to J. Kaczorowski, A. Perelli, "On the structure of the Selberg class, VII: $1\lt d\lt2$", Ann. of Math. (2) 173 (2011), 13971441. Note that, in fact, Kaczorowski and Perelli prove their results for functions in the socalled extended Selberg class $\mathcal S^\sharp$, whose elements are not required to satisfy the Ramanujan hypothesis and the Euler product property. 

