Properties

Label 2-115920-1.1-c1-0-131
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 + 4·11-s + 2·13-s + 2·17-s + 23-s + 25-s − 6·29-s + 8·31-s − 35-s + 10·37-s + 2·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s + 4·55-s − 4·59-s + 10·61-s + 2·65-s + 12·67-s − 12·71-s − 10·73-s − 4·77-s − 16·79-s + 2·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.169·35-s + 1.64·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s − 1.42·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s + 0.216·85-s − 0.635·89-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: $\chi_{115920} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.86386543039810, −13.28655360115611, −12.92956171085150, −12.52510204253964, −11.76545743841819, −11.40264535937627, −11.16072154709972, −10.19145917202096, −9.980719727943913, −9.472501279792635, −8.999171006805200, −8.493077482136310, −7.947607132950040, −7.344349129371497, −6.738989686160744, −6.268694247664333, −5.960932826766222, −5.309004491113752, −4.594676858680593, −4.099107324772375, −3.502695003382917, −2.945064512605412, −2.295273286551247, −1.368834450310167, −1.125539683483342, 0, 1.125539683483342, 1.368834450310167, 2.295273286551247, 2.945064512605412, 3.502695003382917, 4.099107324772375, 4.594676858680593, 5.309004491113752, 5.960932826766222, 6.268694247664333, 6.738989686160744, 7.344349129371497, 7.947607132950040, 8.493077482136310, 8.999171006805200, 9.472501279792635, 9.980719727943913, 10.19145917202096, 11.16072154709972, 11.40264535937627, 11.76545743841819, 12.52510204253964, 12.92956171085150, 13.28655360115611, 13.86386543039810

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