Properties

Label 2-115920-1.1-c1-0-108
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 + 4·11-s + 23-s + 25-s − 2·29-s + 4·31-s − 35-s + 2·37-s − 4·43-s − 8·47-s + 49-s − 2·53-s − 4·55-s − 10·59-s − 12·61-s + 4·67-s − 2·73-s + 4·77-s + 2·79-s + 14·83-s + 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.20·11-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 − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s − 1.30·59-s − 1.53·61-s + 0.488·67-s − 0.234·73-s + 0.455·77-s + 0.225·79-s + 1.53·83-s + 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.79185081075818, −13.47303953398619, −12.83674486139525, −12.16675053197130, −12.04033208237107, −11.38544168080339, −11.05255050903205, −10.55247758034820, −9.812298526920975, −9.475948925506163, −8.893577306265709, −8.450286632171527, −7.899186737550057, −7.428051774670369, −6.881288948668822, −6.215140030029347, −6.046453637603531, −4.943310761668400, −4.794238129988509, −4.095057562462538, −3.508201817030316, −3.064530604162012, −2.178700289291983, −1.529746221625468, −0.9439102523328816, 0, 0.9439102523328816, 1.529746221625468, 2.178700289291983, 3.064530604162012, 3.508201817030316, 4.095057562462538, 4.794238129988509, 4.943310761668400, 6.046453637603531, 6.215140030029347, 6.881288948668822, 7.428051774670369, 7.899186737550057, 8.450286632171527, 8.893577306265709, 9.475948925506163, 9.812298526920975, 10.55247758034820, 11.05255050903205, 11.38544168080339, 12.04033208237107, 12.16675053197130, 12.83674486139525, 13.47303953398619, 13.79185081075818

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