The degree conjecture for the extended Selberg class

Let ${{\mathfrak A}}^{\#}_d$ denote the set of functions $F(s)$ in the extended Selberg class ${{\mathfrak A}}^{\#}$ whose degree is $d_F=d$, where $d_F$ is the degree determined by the functional equation of $F(s)$. The degree conjecture. If $d\geq 0$ is not an integer, then

$${{\mathfrak A}}^{\#}_d=\emptyset.$$

This extends the degree conjecture for the Selberg class to the larger class ${{\mathfrak A}}^{\#}$. The preceding results establish the classification in degree zero and show that the classes of degrees strictly between zero and one are empty; the conjecture concerns all nonintegral nonnegative degrees.