Properties

Label 2-115920-1.1-c1-0-123
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 + 4·13-s + 6·17-s − 4·19-s + 23-s + 25-s − 4·29-s + 8·31-s + 35-s + 2·37-s + 2·41-s + 49-s − 8·53-s − 4·55-s − 10·59-s − 8·61-s + 4·65-s + 4·67-s + 8·71-s − 4·77-s − 6·79-s − 6·83-s + 6·85-s + 14·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.742·29-s + 1.43·31-s + 0.169·35-s + 0.328·37-s + 0.312·41-s + 1/7·49-s − 1.09·53-s − 0.539·55-s − 1.30·59-s − 1.02·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.455·77-s − 0.675·79-s − 0.658·83-s + 0.650·85-s + 1.48·89-s + 0.419·91-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.97983865291270, −13.34243388443417, −12.87351027059653, −12.52486998332888, −11.95239647804729, −11.28098101741750, −10.91323122742555, −10.42989576739275, −10.07536948722346, −9.426976113518724, −8.970769083938215, −8.278861369517176, −7.901222530063345, −7.658850210765258, −6.709842511866772, −6.313016639657957, −5.691807928533729, −5.380081838301260, −4.682734918379023, −4.190542950209872, −3.387848846986339, −2.923316622784501, −2.286448396588790, −1.524321572448875, −1.002868621252759, 0, 1.002868621252759, 1.524321572448875, 2.286448396588790, 2.923316622784501, 3.387848846986339, 4.190542950209872, 4.682734918379023, 5.380081838301260, 5.691807928533729, 6.313016639657957, 6.709842511866772, 7.658850210765258, 7.901222530063345, 8.278861369517176, 8.970769083938215, 9.426976113518724, 10.07536948722346, 10.42989576739275, 10.91323122742555, 11.28098101741750, 11.95239647804729, 12.52486998332888, 12.87351027059653, 13.34243388443417, 13.97983865291270

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