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Properties

Label 2-115920-1.1-c1-0-136
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 − 2·11-s − 2·17-s − 4·19-s − 23-s + 25-s − 4·29-s − 4·31-s − 35-s − 2·37-s − 6·41-s − 2·43-s + 49-s − 10·53-s + 2·55-s − 8·59-s − 14·61-s − 10·67-s + 2·71-s − 2·73-s − 2·77-s − 16·79-s + 12·83-s + 2·85-s + 4·95-s − 4·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.603·11-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.742·29-s − 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 1.04·59-s − 1.79·61-s − 1.22·67-s + 0.237·71-s − 0.234·73-s − 0.227·77-s − 1.80·79-s + 1.31·83-s + 0.216·85-s + 0.410·95-s − 0.406·97-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.06060100992732, −13.54312219592751, −13.15032060874750, −12.56584348817495, −12.23780281932379, −11.62963861808992, −11.08069220329977, −10.79704887446243, −10.30693759267119, −9.716195267971834, −9.008409026380515, −8.762762578109837, −8.095896826077114, −7.670045342239526, −7.264295409220920, −6.558312900964600, −6.101688204901487, −5.486708660160335, −4.817004989282711, −4.510225453590008, −3.809493080067166, −3.241509843470756, −2.605074319569660, −1.851650143229230, −1.403555582479158, 0, 0, 1.403555582479158, 1.851650143229230, 2.605074319569660, 3.241509843470756, 3.809493080067166, 4.510225453590008, 4.817004989282711, 5.486708660160335, 6.101688204901487, 6.558312900964600, 7.264295409220920, 7.670045342239526, 8.095896826077114, 8.762762578109837, 9.008409026380515, 9.716195267971834, 10.30693759267119, 10.79704887446243, 11.08069220329977, 11.62963861808992, 12.23780281932379, 12.56584348817495, 13.15032060874750, 13.54312219592751, 14.06060100992732

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