Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
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Elliptic curves with same $a_p$ for every $p$ are isogenous
If two curves E,F satisfy $\#E(\mathbb{F}_p) = \#F(\mathbb{F}_p)$ for each large prime then E and F are isogenous (conversely, two isogenous curves must have the same values of $\#E(F_p)$ for every $p$...
- 1,043
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If the analytic rank is one then the sign in the functional equation is -1?
Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $L(E, s)$ denote its $L$-function over $\mathbf{Q}$. Also $f$ denotes the weight two cusp form associated to $E$, but this shouldn't be ...
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If $χ_p$ is ramified then $χ(p) = 0$
Consider suppose that $χ_{idelic}$ is the idelic lift of the Dirichlet character $χ$. In my text, it says that if the local character $χ_p$ is ramified that is there exists some u
$\in Z_p^{x}$ such ...
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Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions
Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e.,
$$
\zeta_K(s)=\prod_{\...
- 787
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Proposition 16.5.4 in Ireland-Rosen
We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
- 4,429
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Reconciling different ideal-theoretic definitions of Hecke Characters
I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as:
Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
- 787
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M/V Multiplicative NT : Theorem 11.3 and the Siegel zero
Two questions regarding Theorem 11.3 in the book of Montgomery & Vaughan Multiplicative Number Theory on the section "Case 4. Quadratic $χ$, real zeros.":
First the book supposes there ...
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Why $L(s, χ)$ is nonzero for $s$ real and $χ$ complex?
A quick question...
In Section 11.1 of the book of Montgomery & Vaughan's Multiplicative Number Theory when studying the case $χ$ complex it doesn't suppose there can be a real zero for $L(s, χ)$ ...
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An estimate on GRH for symmetric power L-function
I have a question in an article of K. Soundararajan and Matthew P. Young : The second moment of quadratic twists of
modular L-functions.
In their article, one says that using the GRH for $L\left(s, \...
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Functional equation of the Hecke L function in ideal term and "ideal number" term (Neukirch Chapter VII)
$\def\A{\mathbb{A}}
\def\B{\mathbb{B}}
\def\C{\mathbb{C}}
\newcommand{\Cx}{\mathbb{C}^{\times}}
\def\F{\mathbb{F}}
\def\G{\mathbb{G}}
\def\H{\mathbb{H}}
\def\K{\mathbb{K}}
\def\M{\mathbb{M}}
\def\N{\...
- 1,016
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Some curios sums of Hurwitz zeta-function and Lehmer's totient problem [closed]
For all squarefree $k\in \mathbb N$
$$\left|\frac{\sum_{n=1}^{k-1}\zeta_H(-1,n/k)}{\sum_{n=1}^{k-1}\chi_0(n)\zeta_H(-1,n/k)}\right|=\left|\frac{\sum_{n=1}^{k-1}1}{\sum_{n=1}^{k-1}\chi_0(n)}\right|=\...
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Explicit translation of Ramanujan’s condition for the Selberg Class?
According to many sites such as Wikipedia, the Ramanujan condition of the Selberg class can be stated as
$$\boxed{\forall\epsilon>0:a_n\ll_\epsilon n^\epsilon}$$
My question is: what exactly does ...
- 310
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q-series Expansion at Cusp and L-Functions
We know that the coefficients at $\infty$ of a modular form that is an eigenfunction of all Hecke operators in, say, $\Gamma_0(q)$, give an $L$-function with an Euler product and a functional equation....
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Zeros of L function on the 0.5 line
Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$?
Is there an upper bound for this count when we count ...
- 1,132
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A Conjecture Relating Modulo Arithmetic and the Riemann Zeta Function.
I recently created a function that has perplexed many of my fellow amateur mathematicians. It goes something like this: $$f\left(g(x)\right)=\frac{1}{N^{2}}\sum_{n=1}^{N}\left(Ng(x)\operatorname{mod}n\...
