Properties

Label 2-115920-1.1-c1-0-119
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 + 6·11-s − 2·13-s − 6·19-s − 23-s + 25-s + 4·29-s − 2·31-s + 35-s − 6·37-s − 6·41-s − 6·43-s + 2·47-s + 49-s + 2·53-s + 6·55-s + 4·59-s − 2·61-s − 2·65-s + 2·67-s + 10·71-s − 6·73-s + 6·77-s − 6·83-s − 10·89-s − 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.80·11-s − 0.554·13-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.742·29-s − 0.359·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s − 0.914·43-s + 0.291·47-s + 1/7·49-s + 0.274·53-s + 0.809·55-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + 0.244·67-s + 1.18·71-s − 0.702·73-s + 0.683·77-s − 0.658·83-s − 1.05·89-s − 0.209·91-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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.87632111703539, −13.48952203134021, −12.79018370766648, −12.31425402791057, −11.99403718215599, −11.43698518962249, −10.99362979985532, −10.35273231248352, −9.953026712423465, −9.475066313579985, −8.809752394511699, −8.572180144603959, −8.078067035712978, −7.060529169435856, −6.971189226004025, −6.345890857514572, −5.897004727628081, −5.173604952326838, −4.667538098811424, −4.096934966829665, −3.627944478669822, −2.887161141972489, −2.045913718263965, −1.721289881614764, −0.9789578358016160, 0, 0.9789578358016160, 1.721289881614764, 2.045913718263965, 2.887161141972489, 3.627944478669822, 4.096934966829665, 4.667538098811424, 5.173604952326838, 5.897004727628081, 6.345890857514572, 6.971189226004025, 7.060529169435856, 8.078067035712978, 8.572180144603959, 8.809752394511699, 9.475066313579985, 9.953026712423465, 10.35273231248352, 10.99362979985532, 11.43698518962249, 11.99403718215599, 12.31425402791057, 12.79018370766648, 13.48952203134021, 13.87632111703539

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