Properties

Label 2-8280-1.1-c1-0-4
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2.73·7-s − 5.97·11-s − 5.40·13-s − 6.94·17-s − 3.97·19-s + 23-s + 25-s + 9.29·29-s − 3.54·31-s − 2.73·35-s + 7.17·37-s − 10.1·41-s + 1.42·43-s − 10.4·47-s + 0.487·49-s + 4.04·53-s − 5.97·55-s + 8.13·59-s − 3.34·61-s − 5.40·65-s − 1.70·67-s − 7.56·71-s + 7.85·73-s + 16.3·77-s − 1.36·79-s + 13.9·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.03·7-s − 1.80·11-s − 1.49·13-s − 1.68·17-s − 0.912·19-s + 0.208·23-s + 0.200·25-s + 1.72·29-s − 0.636·31-s − 0.462·35-s + 1.18·37-s − 1.59·41-s + 0.217·43-s − 1.51·47-s + 0.0696·49-s + 0.556·53-s − 0.805·55-s + 1.05·59-s − 0.428·61-s − 0.669·65-s − 0.208·67-s − 0.897·71-s + 0.919·73-s + 1.86·77-s − 0.153·79-s + 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5232175333\)
\(L(\frac12)\) \(\approx\) \(0.5232175333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 5.97T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 + 3.97T + 19T^{2} \)
29 \( 1 - 9.29T + 29T^{2} \)
31 \( 1 + 3.54T + 31T^{2} \)
37 \( 1 - 7.17T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 1.42T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 4.04T + 53T^{2} \)
59 \( 1 - 8.13T + 59T^{2} \)
61 \( 1 + 3.34T + 61T^{2} \)
67 \( 1 + 1.70T + 67T^{2} \)
71 \( 1 + 7.56T + 71T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 + 1.36T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 + 6.49T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82602049399249134707841913227, −6.90202389469066873898691023300, −6.60407959943068444048070420442, −5.75174000354469733688254978686, −4.86578433825933726133727548798, −4.58033463405559768853368965771, −3.26102459286128709439405811516, −2.54422616734848551979751937159, −2.13272666985618256259963074159, −0.32240760495662175513553040452, 0.32240760495662175513553040452, 2.13272666985618256259963074159, 2.54422616734848551979751937159, 3.26102459286128709439405811516, 4.58033463405559768853368965771, 4.86578433825933726133727548798, 5.75174000354469733688254978686, 6.60407959943068444048070420442, 6.90202389469066873898691023300, 7.82602049399249134707841913227

Graph of the $Z$-function along the critical line