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Properties

Label 2-115920-1.1-c1-0-83
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 − 3·11-s + 2·17-s − 7·19-s + 23-s + 25-s + 2·29-s + 4·31-s − 35-s + 2·37-s − 3·41-s − 10·43-s − 47-s + 49-s + 3·53-s − 3·55-s − 5·59-s + 3·61-s + 14·67-s + 2·73-s + 3·77-s + 6·79-s − 8·83-s + 2·85-s + 12·89-s − 7·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.485·17-s − 1.60·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.468·41-s − 1.52·43-s − 0.145·47-s + 1/7·49-s + 0.412·53-s − 0.404·55-s − 0.650·59-s + 0.384·61-s + 1.71·67-s + 0.234·73-s + 0.341·77-s + 0.675·79-s − 0.878·83-s + 0.216·85-s + 1.27·89-s − 0.718·95-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.63409775051937, −13.34293265789170, −13.01254640464284, −12.34334204628415, −12.11953311343496, −11.33833105061286, −10.81154429407653, −10.46096069720300, −9.882871175385528, −9.650730961144362, −8.873039087934079, −8.299187384169187, −8.138951618913162, −7.354298035561568, −6.662452552840262, −6.470972570504083, −5.778880651900502, −5.228548918885110, −4.765663669235370, −4.122787560549442, −3.446896951192328, −2.838651452427814, −2.305359518197657, −1.701303527346311, −0.7941783550420782, 0, 0.7941783550420782, 1.701303527346311, 2.305359518197657, 2.838651452427814, 3.446896951192328, 4.122787560549442, 4.765663669235370, 5.228548918885110, 5.778880651900502, 6.470972570504083, 6.662452552840262, 7.354298035561568, 8.138951618913162, 8.299187384169187, 8.873039087934079, 9.650730961144362, 9.882871175385528, 10.46096069720300, 10.81154429407653, 11.33833105061286, 12.11953311343496, 12.34334204628415, 13.01254640464284, 13.34293265789170, 13.63409775051937

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