We will introduce two new classes of Dirichlet series which are monoids under multiplication. The first class $\mathfrak{ A}^{ \#}$ contains both the extended Selberg class $\mathscr{ S}^{ \#}$ of Kaczorowski and Perelli as well as many $L$-functions attached to automorphic representations of ${ \rm GL}_n({ \mathbb A}_K)$, where ${ \mathbb A}_K$ denotes the ad\`eles over the number field $K$ (these representations need not be unitary or generic). This is in contrast to the class $\mathscr{ S}^{ \#}$ which is smaller and is known to contain, very few of these $L$-functions. The larger class is obtained by weakening the requirement for absolute convergence, allowing a finite number of poles, allowing more general gamma factors and by allowing the series to have trivial zeros to the right of $\mathrm{ Re}(s)=1/2$, while retaining the other axioms of the extended Selberg class. We will classify series in $\mathfrak{ A}^{ \#}$ of degree $d$ when $d\le 1$ (when $d=1$, we will assume absolute convergence in $\mathrm{ Re}(s)>1$). We will further prove a primitivity result for the $L$-functions of cuspidal eigenforms on ${ \rm GL}_2({ \mathbb A}_{ \mathbb Q})$ and a theorem allowing us to compare the zeros of tensor product $L$-functions of ${ \rm GL}_n({ \mathbb A}_K)$ which cannot be deduced from previous classification results. The second class $\mathfrak{ G}^{ \#}\subset\mathfrak{ A}^{ \#}$, which also contains $\mathscr{ S}^{ \#}$, more closely models the behaviour of $L$-functions of unitary globally generic representations of ${ \rm GL}_n({ \mathbb A}_K)$.
Categories: math.NT