Properties

Label 2-8280-1.1-c1-0-38
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 − 0.761·7-s + 2.86·11-s + 0.864·13-s − 0.761·17-s + 6.38·19-s − 23-s + 25-s + 1.62·29-s − 3.62·31-s − 0.761·35-s + 2.76·37-s + 7.62·41-s + 1.72·43-s + 10.3·47-s − 6.42·49-s − 0.761·53-s + 2.86·55-s − 3.35·59-s − 2.11·61-s + 0.864·65-s − 7.74·67-s − 13.9·71-s + 4.92·73-s − 2.18·77-s + 10.5·79-s − 0.490·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.287·7-s + 0.863·11-s + 0.239·13-s − 0.184·17-s + 1.46·19-s − 0.208·23-s + 0.200·25-s + 0.301·29-s − 0.651·31-s − 0.128·35-s + 0.453·37-s + 1.19·41-s + 0.263·43-s + 1.51·47-s − 0.917·49-s − 0.104·53-s + 0.386·55-s − 0.436·59-s − 0.271·61-s + 0.107·65-s − 0.945·67-s − 1.65·71-s + 0.576·73-s − 0.248·77-s + 1.18·79-s − 0.0538·83-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

\(L(1)\) \(\approx\) \(2.488632529\)
\(L(\frac12)\) \(\approx\) \(2.488632529\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.761T + 7T^{2} \)
11 \( 1 - 2.86T + 11T^{2} \)
13 \( 1 - 0.864T + 13T^{2} \)
17 \( 1 + 0.761T + 17T^{2} \)
19 \( 1 - 6.38T + 19T^{2} \)
29 \( 1 - 1.62T + 29T^{2} \)
31 \( 1 + 3.62T + 31T^{2} \)
37 \( 1 - 2.76T + 37T^{2} \)
41 \( 1 - 7.62T + 41T^{2} \)
43 \( 1 - 1.72T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 0.761T + 53T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 + 2.11T + 61T^{2} \)
67 \( 1 + 7.74T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 4.92T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 0.490T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 6.77T + 97T^{2} \)
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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

−7.57678701057787920426637640174, −7.25743275812406750513092805871, −6.21551943475401613350253780204, −5.95363595535186813464035634779, −5.03795435440640961664513496202, −4.24875028426973949862113138864, −3.46596337953633871809561429491, −2.71559181380713158830666389591, −1.68497910534793668716666021733, −0.814168308054472392000466319306, 0.814168308054472392000466319306, 1.68497910534793668716666021733, 2.71559181380713158830666389591, 3.46596337953633871809561429491, 4.24875028426973949862113138864, 5.03795435440640961664513496202, 5.95363595535186813464035634779, 6.21551943475401613350253780204, 7.25743275812406750513092805871, 7.57678701057787920426637640174

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