Properties

Label 2-8280-1.1-c1-0-8
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2.94·7-s + 4.49·11-s − 6.14·13-s − 7.63·17-s + 6.67·19-s − 23-s + 25-s − 7.87·29-s − 7.74·31-s + 2.94·35-s − 1.41·37-s − 5.34·41-s + 4.46·43-s − 0.681·47-s + 1.65·49-s + 6.97·53-s − 4.49·55-s + 8.08·59-s + 13.3·61-s + 6.14·65-s + 12.6·67-s + 2.37·71-s − 13.0·73-s − 13.2·77-s − 4.67·79-s + 3.36·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.11·7-s + 1.35·11-s − 1.70·13-s − 1.85·17-s + 1.53·19-s − 0.208·23-s + 0.200·25-s − 1.46·29-s − 1.39·31-s + 0.497·35-s − 0.232·37-s − 0.835·41-s + 0.681·43-s − 0.0994·47-s + 0.236·49-s + 0.958·53-s − 0.606·55-s + 1.05·59-s + 1.70·61-s + 0.762·65-s + 1.54·67-s + 0.282·71-s − 1.53·73-s − 1.50·77-s − 0.526·79-s + 0.369·83-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

\(L(1)\) \(\approx\) \(0.8608984835\)
\(L(\frac12)\) \(\approx\) \(0.8608984835\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.94T + 7T^{2} \)
11 \( 1 - 4.49T + 11T^{2} \)
13 \( 1 + 6.14T + 13T^{2} \)
17 \( 1 + 7.63T + 17T^{2} \)
19 \( 1 - 6.67T + 19T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 + 7.74T + 31T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + 5.34T + 41T^{2} \)
43 \( 1 - 4.46T + 43T^{2} \)
47 \( 1 + 0.681T + 47T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 - 8.08T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 2.37T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 4.67T + 79T^{2} \)
83 \( 1 - 3.36T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 6.91T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.54602235567401951535865358759, −6.99124683775725221986110216125, −6.75703226700936237892963878158, −5.69308681192335317780951559387, −5.05437586032987531828927250336, −4.04860553574221875328530049411, −3.66286049315720303181827446632, −2.68313812389255078473513518096, −1.85521080649305126421985330562, −0.43575992711945093142150898099, 0.43575992711945093142150898099, 1.85521080649305126421985330562, 2.68313812389255078473513518096, 3.66286049315720303181827446632, 4.04860553574221875328530049411, 5.05437586032987531828927250336, 5.69308681192335317780951559387, 6.75703226700936237892963878158, 6.99124683775725221986110216125, 7.54602235567401951535865358759

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