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Properties

Label 2-115920-1.1-c1-0-130
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 − 11-s + 4·13-s − 19-s + 23-s + 25-s + 8·29-s − 4·31-s + 35-s + 8·37-s − 7·41-s + 6·43-s − 3·47-s + 49-s − 5·53-s − 55-s + 11·59-s + 13·61-s + 4·65-s − 14·67-s + 8·71-s − 77-s − 12·79-s − 6·83-s − 4·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s − 0.229·19-s + 0.208·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s + 0.169·35-s + 1.31·37-s − 1.09·41-s + 0.914·43-s − 0.437·47-s + 1/7·49-s − 0.686·53-s − 0.134·55-s + 1.43·59-s + 1.66·61-s + 0.496·65-s − 1.71·67-s + 0.949·71-s − 0.113·77-s − 1.35·79-s − 0.658·83-s − 0.423·89-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: 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 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.89173073682810, −13.23470595239241, −13.04843604830776, −12.47060130285249, −11.85569306664365, −11.30924155796395, −11.02546551024187, −10.37930287343646, −10.05354458118303, −9.433029668414649, −8.875036411642423, −8.392890564573288, −8.075623436977664, −7.388352373192197, −6.721209480127925, −6.408207041822594, −5.702600114119191, −5.346019151965237, −4.678047378278945, −4.107887280003259, −3.571490936076725, −2.740904990765426, −2.396593423213442, −1.408497369273495, −1.102114100404189, 0, 1.102114100404189, 1.408497369273495, 2.396593423213442, 2.740904990765426, 3.571490936076725, 4.107887280003259, 4.678047378278945, 5.346019151965237, 5.702600114119191, 6.408207041822594, 6.721209480127925, 7.388352373192197, 8.075623436977664, 8.392890564573288, 8.875036411642423, 9.433029668414649, 10.05354458118303, 10.37930287343646, 11.02546551024187, 11.30924155796395, 11.85569306664365, 12.47060130285249, 13.04843604830776, 13.23470595239241, 13.89173073682810

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