# Questions tagged [selberg-class]

Questions about Selberg class and the related conjectures such as the analogue of Riemann Hypothesis, Selberg's orthonormality conjecture, degree conjecture, general converse conjecture that says the Selberg class exactly consists of automorphic L-functions, etc.

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### Selberg class definition and Riemann hypothesis

Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph: "The condition that the real part of $\mu_i$ be non-negative is because ...
1 vote
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### Does the Ramanujan-Petersson condition correspond to a Fourier type property?

The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
171 views

### Does Rankin-Selberg convolution preserve primitivity?

Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
123 views

### Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series $$L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$ satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
159 views

### Does this particular L-series built from L-functions of prime degree define an L-function?

Throughout this question, I call 'L-function' any automorphic L-function belonging to the Selberg class. Suppose $(F_i)_{(i>0)}$ is a sequence of L-functions with $F_i$ of degree $p_i$ ...
238 views

### Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function

Calling '$L$-function' any automorphic $L$-function belonging to the Selberg class, what are the known $L$-functions $L(s,F)$ and $L(s,G)$ of respective degrees $d$ and $d'$ such that the Rankin-...
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### seminar about the strong multiplicity one for the Selberg class

Very recently, a seminar took place in Seoul with Haseo Ki as an invited speaker to talk about the strong multiplicity one theorem for the whole Selberg class that he did manage to prove. I would like ...
671 views

### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...
228 views

### Tensor product of two elements of the Selberg class

Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is ...
I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...