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Properties

Label 2-115920-1.1-c1-0-24
Degree $2$
Conductor $115920$
Sign $-1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 6·11-s − 13-s − 6·17-s − 2·19-s − 23-s + 25-s − 9·29-s + 31-s + 35-s + 8·37-s − 9·41-s − 2·43-s − 3·47-s + 49-s + 6·55-s + 12·59-s + 8·61-s + 65-s − 8·67-s + 9·71-s + 5·73-s + 6·77-s − 14·79-s + 6·83-s + 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.67·29-s + 0.179·31-s + 0.169·35-s + 1.31·37-s − 1.40·41-s − 0.304·43-s − 0.437·47-s + 1/7·49-s + 0.809·55-s + 1.56·59-s + 1.02·61-s + 0.124·65-s − 0.977·67-s + 1.06·71-s + 0.585·73-s + 0.683·77-s − 1.57·79-s + 0.658·83-s + 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.46094023535042, −13.38537316369736, −12.94352802574298, −12.59338092239280, −11.79760395004162, −11.41223928225117, −10.91363152595391, −10.51909751707363, −9.927845406196946, −9.550681389934870, −8.841151683569304, −8.351657457587389, −7.976419333705176, −7.372583595864030, −6.938135743457024, −6.379846664270124, −5.727771953592111, −5.185971124658781, −4.736185109474990, −4.076515306523316, −3.563338767211411, −2.784095824093164, −2.348473023354065, −1.792488494572024, −0.5388973832348525, 0, 0.5388973832348525, 1.792488494572024, 2.348473023354065, 2.784095824093164, 3.563338767211411, 4.076515306523316, 4.736185109474990, 5.185971124658781, 5.727771953592111, 6.379846664270124, 6.938135743457024, 7.372583595864030, 7.976419333705176, 8.351657457587389, 8.841151683569304, 9.550681389934870, 9.927845406196946, 10.51909751707363, 10.91363152595391, 11.41223928225117, 11.79760395004162, 12.59338092239280, 12.94352802574298, 13.38537316369736, 13.46094023535042

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