Properties

Label 2-8280-1.1-c1-0-0
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 − 5.18·7-s − 5.53·11-s + 1.78·13-s + 2.22·17-s − 8.22·19-s − 23-s + 25-s − 7.06·29-s − 2.06·31-s + 5.18·35-s − 5.02·37-s − 11.4·41-s − 8.62·43-s − 1.37·47-s + 19.9·49-s + 4.46·53-s + 5.53·55-s − 11.1·59-s + 7.81·61-s − 1.78·65-s − 1.96·67-s + 8.51·71-s − 10.7·73-s + 28.6·77-s − 14.0·79-s − 6.83·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.96·7-s − 1.66·11-s + 0.496·13-s + 0.539·17-s − 1.88·19-s − 0.208·23-s + 0.200·25-s − 1.31·29-s − 0.371·31-s + 0.876·35-s − 0.826·37-s − 1.79·41-s − 1.31·43-s − 0.200·47-s + 2.84·49-s + 0.613·53-s + 0.745·55-s − 1.45·59-s + 1.00·61-s − 0.221·65-s − 0.240·67-s + 1.01·71-s − 1.25·73-s + 3.26·77-s − 1.57·79-s − 0.750·83-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

\(L(1)\) \(\approx\) \(0.05780330460\)
\(L(\frac12)\) \(\approx\) \(0.05780330460\)
\(L(\frac{3}{2})\) 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 \( 1 + 5.18T + 7T^{2} \)
11 \( 1 + 5.53T + 11T^{2} \)
13 \( 1 - 1.78T + 13T^{2} \)
17 \( 1 - 2.22T + 17T^{2} \)
19 \( 1 + 8.22T + 19T^{2} \)
29 \( 1 + 7.06T + 29T^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
37 \( 1 + 5.02T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 + 8.62T + 43T^{2} \)
47 \( 1 + 1.37T + 47T^{2} \)
53 \( 1 - 4.46T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 7.81T + 61T^{2} \)
67 \( 1 + 1.96T + 67T^{2} \)
71 \( 1 - 8.51T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 6.83T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + 4.19T + 97T^{2} \)
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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

−7.79718138311057409096675919620, −7.02144087933824976575810742517, −6.50106794601717640382071702906, −5.77418269124391140245653410707, −5.14795285629342753355388132269, −4.05597544588019902193498185721, −3.44480098219122757321893823108, −2.84913517630237701829382019422, −1.89312989709893550645436675065, −0.10897282743110876172984305429, 0.10897282743110876172984305429, 1.89312989709893550645436675065, 2.84913517630237701829382019422, 3.44480098219122757321893823108, 4.05597544588019902193498185721, 5.14795285629342753355388132269, 5.77418269124391140245653410707, 6.50106794601717640382071702906, 7.02144087933824976575810742517, 7.79718138311057409096675919620

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