All Questions
Tagged with dirichlet-series l-functions
21
questions
2
votes
1
answer
113
views
Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
6
votes
1
answer
146
views
Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
- 173
1
vote
1
answer
114
views
Looking for table of special values of the Dirichlet $L$-function
For double checking calculations I made I'd like to find a table of values of $L(-1,\chi_D)$ for small positive fundamental discriminats $D$. It there a table somewhere in the internet? Where?
With $\...
- 6,184
1
vote
1
answer
226
views
$L$-function of elliptic curves expansion into Dirichlet series
Let $E/\mathbb{Q}$ be an elliptic curve. The $L$-function of $E$ is defined to be the Euler product
$$
L_E(s) = \prod_{\text{ bad }p} (1 - a_p p^{-s})^{-1} \prod_{\text{ good }p} (1 - a_p p^{-s} + p^{...
- 1,246
0
votes
0
answers
217
views
Dirichlet L-series and Hecke L-series
I'm working on L-series (reading Rosen's book Number Theory in Function fields) and i read that Dirichlet $L$-series are supposed to be a special case of Hecke $L$-series, and i can't understand why ?
- 1
0
votes
1
answer
61
views
An $L-$function and a $J-$function. Related?
Consider a Dirichlet series for a non real character of modulus $q$
$$ L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} $$
and $s\in\Bbb C$ with real part greater than one.
Consider a $J$-series $$ J(s,...
- 756
1
vote
0
answers
156
views
Non-vanishing of Dirichlet $L$-function $L(s,\chi)$ for $\Re(s)=1$ [duplicate]
I know that if $\chi$ is a non-principal Dirichlet character then the $L$-function $L(s,\chi)$ doesn't vanish for $s=1$. But, how about $s=1+it$ with $t\neq 0$? I found in this post: Zeros of ...
- 299
2
votes
0
answers
53
views
Do you know about the textbook of Selberg class of Dirichlet series?
I have read the Atle Selberg's thesis named "Old and new conjectures and results about a class of Dirichlet series". At the end of this thesis, he wrote
"A more complete account with proofs is ...
- 101
0
votes
0
answers
81
views
uniform convergence of $L(s,\chi)$ for $\Re(s) ≥ 1 + \delta$ " due to absolute convergence for $\Re(s)>1$?
On page 6 of this link, lemma 2.4 shows $L(s,\chi)$ is absolutely convergent for $\Re(s)>1$.
I understand the proof. However,
they also add: "The above proof also shows that for any $\delta > ...
- 1,309
1
vote
1
answer
224
views
Euler product for $ \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 $
I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product:
$$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2
= q \prod_{n=1}^\infty (1 - q^n)...
- 24.5k
1
vote
1
answer
978
views
Bounds for Dirichlet L-functions
In the half plane $\sigma$=Re(s) > 1 , one can find bounds for the Riemann zeta function $\zeta$(s) using either its convergent series or product formula.$\,$ From the Dirichlet series we get the ...
- 2,437
0
votes
0
answers
203
views
Question regarding the number of zeros of Dirichlet L-function
I have encountered the following result: Let $T\geq 2$, and let $N^*(\alpha, q, T)$ denote the number of zeros of all the L-functions $L(s, \chi)$ with primitive characters $\chi$ modulo $q$ in the ...
- 2,913
5
votes
0
answers
570
views
Functional equation of the complete $L$-function of the twisted $L$-function of a cuspidal modular form
Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the ...
3
votes
1
answer
79
views
Determine Fourier coefficients by the values of its L-Series.
suppose I know the values of $\sum_{n=1}^\infty \frac{a_n}{n^k}$ for all $k=1,2,...$. Is there a way/tool to determine the coefficients $a_n$ from this (which might not be unique)?
I would appreciate ...
- 115
4
votes
1
answer
875
views
Showing $L(1,\chi)$ is positive given that it's nonzero
Let me first provide context for this question.
There is a series of four exercises in Ireland & Rosen's book (in second edition it's exercises 14-17 in chaprer 16), aim of which is (although ...
- 26.8k