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Properties

Label 2-115920-1.1-c1-0-137
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 4·11-s − 2·13-s − 6·17-s − 4·19-s + 23-s + 25-s + 2·29-s − 4·31-s + 35-s − 10·37-s − 10·41-s − 12·43-s − 12·47-s + 49-s − 14·53-s − 4·55-s + 8·59-s + 10·61-s − 2·65-s + 4·67-s + 8·71-s − 6·73-s − 4·77-s + 12·83-s − 6·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.169·35-s − 1.64·37-s − 1.56·41-s − 1.82·43-s − 1.75·47-s + 1/7·49-s − 1.92·53-s − 0.539·55-s + 1.04·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s − 0.455·77-s + 1.31·83-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: \(2\)
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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.03015092976958, −13.51031057046586, −13.06942436541768, −12.83941663198725, −12.21890651395330, −11.52230206028192, −11.21223536114185, −10.63125072504495, −10.18561811687056, −9.858319315208649, −9.143524408680842, −8.457488832774880, −8.405865462416205, −7.722251738341274, −6.861657730823745, −6.782981062677346, −6.140871002360892, −5.278905797956126, −4.865510266040434, −4.793750267650331, −3.668512236658764, −3.272525291379897, −2.304956677424423, −2.115384205551026, −1.430442661426121, 0, 0, 1.430442661426121, 2.115384205551026, 2.304956677424423, 3.272525291379897, 3.668512236658764, 4.793750267650331, 4.865510266040434, 5.278905797956126, 6.140871002360892, 6.782981062677346, 6.861657730823745, 7.722251738341274, 8.405865462416205, 8.457488832774880, 9.143524408680842, 9.858319315208649, 10.18561811687056, 10.63125072504495, 11.21223536114185, 11.52230206028192, 12.21890651395330, 12.83941663198725, 13.06942436541768, 13.51031057046586, 14.03015092976958

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