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Results tagged with l-functions
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user 43108
Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.
2
votes
what is exactly the difference between the Selberg class and the set of Artin L-functions?
As far as I know, the precise conjecture for what you are asking is:
All the elements of the Selberg class that are not Artin L-functions
are:
motivic L-functions of dimension bigger than 0.
transc …
- 17.6k
2
votes
Does it exist a p-adic L function which interpolates the values of the complex one at positi...
The problem is that your $L_N(k,\xi\chi)$ do not lie in an extension of $\mathbb{Q}_p$.
You can interpolate the algebraic part, but using the functional equation of the original Dirichelt L-functions …
- 17.6k
6
votes
Accepted
Corvallis 1979 proceedings
Given the importance of the proceedings, I leave a link to the copy at Library Genesis:
A. Borel and W. Casselman (Editors) Automorphic Forms, Representations and L-functions (1979)
- 17.6k
7
votes
Irrationality of Dedekind zeta values
As with all motivic L-functions, irrationality of special values breaks into two rather different problems:
Critical values. In the case of Dedekind zeta functions, those are the even positive intege …
- 17.6k
5
votes
Are there L-functions of degree 1 that aren't Hecke L-functions?
The answer is no.
Kaczorowski and Perelli proved the classification of L-functions with degree 1, and the L-functions with that degree turn out to be the Riemann zeta function and Dirichlet L-functio …
- 17.6k
2
votes
1
answer
385
views
Automorphic L-functions over $GL_n( \mathbb{Q} )$
A paper by Kaczorowski & Perelli arXiv:1207.2312 dealing with the elements of the Selber class with degree two suggests that $S_2$ coincides with the automorphic l-functions over $GL_2( \mathbb{Q} )$. …
- 17.6k
9
votes
1
answer
890
views
Tate's thesis for Artin L-functions
As far as I know, Tate's thesis has been successfully applied in two fronts:
Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid)
Automorphic L-functions, by Jacquet, Shalika, Shap …
- 17.6k
5
votes
Relation of these two Dirichlet $L$-functions
It follows from the functorial properties of characters of representations (1-dimensional as in the case of Dirichlet or not) that:
$$L(s,\chi) \times L(s,\psi)=L(s,\chi+\psi)$$
This basically follo …
- 17.6k
2
votes
Accepted
Consequences of the degree conjecture
I think there are no direct consequences of the conjecture, but it seems like the natural first step in solving the bigger orthonormality conjecture (and/or related issues like unique factorization) w …
- 17.6k
5
votes
Accepted
Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function
An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.
This includes two important cases: Hecke characters and holom …
- 17.6k
11
votes
3
answers
1k
views
References for general Hasse-Weil zeta function
Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am lookin …
- 17.6k
13
votes
0
answers
284
views
Propagation of modularity and the Artin conjecture
The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century.
A very natural question is, what …
- 17.6k
13
votes
0
answers
620
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There a …
- 17.6k
16
votes
3
answers
1k
views
First formulation of the Dedekind and Hasse-Weil conjectures
I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
…
- 17.6k
16
votes
1
answer
2k
views
Automorphic factorization of Dedekind zeta functions
It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters di …
- 17.6k