Properties

Label 2-8280-1.1-c1-0-107
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 1.56·7-s + 2·11-s + 0.561·13-s − 5.56·17-s − 2·19-s + 23-s + 25-s − 0.123·29-s − 8.12·31-s + 1.56·35-s − 3.56·37-s + 4.12·41-s − 10.2·43-s − 3.68·47-s − 4.56·49-s − 4.43·53-s + 2·55-s + 5.56·59-s − 9.12·61-s + 0.561·65-s − 11.5·67-s + 5·71-s + 3.43·73-s + 3.12·77-s − 9.12·79-s + 4.68·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.590·7-s + 0.603·11-s + 0.155·13-s − 1.34·17-s − 0.458·19-s + 0.208·23-s + 0.200·25-s − 0.0228·29-s − 1.45·31-s + 0.263·35-s − 0.585·37-s + 0.643·41-s − 1.56·43-s − 0.537·47-s − 0.651·49-s − 0.609·53-s + 0.269·55-s + 0.724·59-s − 1.16·61-s + 0.0696·65-s − 1.41·67-s + 0.593·71-s + 0.402·73-s + 0.355·77-s − 1.02·79-s + 0.514·83-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: \(1\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 + 5.56T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 + 0.123T + 29T^{2} \)
31 \( 1 + 8.12T + 31T^{2} \)
37 \( 1 + 3.56T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 3.68T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 - 5.56T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 - 3.43T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 + 3.12T + 97T^{2} \)
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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

−7.39422581994701323855381367727, −6.70345532552626087413801820057, −6.19679791740728112859349488778, −5.31668119883247553272483166669, −4.67648906820745712549583430050, −3.97155902444509818503704612250, −3.07599016680113969757096304514, −2.01740960175750564163160153787, −1.49416692446185920011985731539, 0, 1.49416692446185920011985731539, 2.01740960175750564163160153787, 3.07599016680113969757096304514, 3.97155902444509818503704612250, 4.67648906820745712549583430050, 5.31668119883247553272483166669, 6.19679791740728112859349488778, 6.70345532552626087413801820057, 7.39422581994701323855381367727

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