Properties

Label 2-115920-1.1-c1-0-109
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 + 2·13-s + 2·17-s + 4·19-s − 23-s + 25-s − 6·29-s − 2·31-s − 35-s − 2·37-s + 6·41-s + 2·43-s + 12·47-s + 49-s − 6·53-s − 8·59-s + 2·61-s − 2·65-s + 14·67-s − 12·71-s + 4·73-s − 6·79-s − 12·83-s − 2·85-s − 6·89-s + 2·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.169·35-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.04·59-s + 0.256·61-s − 0.248·65-s + 1.71·67-s − 1.42·71-s + 0.468·73-s − 0.675·79-s − 1.31·83-s − 0.216·85-s − 0.635·89-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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.95579765508323, −13.27316007986270, −12.94755495649180, −12.21305269813772, −12.04484772645067, −11.30278036153554, −11.02789703255237, −10.59566828182678, −9.838127994793601, −9.493133041789284, −8.855780887948334, −8.478379581668181, −7.789036443160121, −7.417956874686154, −7.105567052391308, −6.154087918078561, −5.827565313538967, −5.254685585645022, −4.684178090284884, −3.962428093607031, −3.650197076910913, −2.946053012559902, −2.288459905978115, −1.473780368998690, −0.9446944494655338, 0, 0.9446944494655338, 1.473780368998690, 2.288459905978115, 2.946053012559902, 3.650197076910913, 3.962428093607031, 4.684178090284884, 5.254685585645022, 5.827565313538967, 6.154087918078561, 7.105567052391308, 7.417956874686154, 7.789036443160121, 8.478379581668181, 8.855780887948334, 9.493133041789284, 9.838127994793601, 10.59566828182678, 11.02789703255237, 11.30278036153554, 12.04484772645067, 12.21305269813772, 12.94755495649180, 13.27316007986270, 13.95579765508323

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