Properties

Label 2-115920-1.1-c1-0-100
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 − 5·11-s + 3·13-s + 5·17-s − 23-s + 25-s − 3·29-s − 6·31-s + 35-s − 4·37-s + 2·43-s − 9·47-s + 49-s + 6·53-s − 5·55-s − 6·59-s + 10·61-s + 3·65-s − 4·67-s − 8·71-s + 10·73-s − 5·77-s + 15·79-s + 12·83-s + 5·85-s + 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.832·13-s + 1.21·17-s − 0.208·23-s + 1/5·25-s − 0.557·29-s − 1.07·31-s + 0.169·35-s − 0.657·37-s + 0.304·43-s − 1.31·47-s + 1/7·49-s + 0.824·53-s − 0.674·55-s − 0.781·59-s + 1.28·61-s + 0.372·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.569·77-s + 1.68·79-s + 1.31·83-s + 0.542·85-s + 1.05·89-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.64777632014016, −13.42296309840853, −12.99375820075330, −12.38158940291498, −12.01744548507564, −11.30533315216121, −10.86599311090632, −10.42129855010952, −10.12215404327801, −9.330799658151005, −9.079845037421382, −8.241314050650861, −7.932132252331186, −7.576927032515096, −6.828016183396789, −6.298927057108586, −5.553612119047229, −5.363135800573263, −4.909254403771076, −3.978110699316804, −3.515785969200492, −2.907158372862065, −2.198427760487026, −1.664130959494908, −0.9116285240348300, 0, 0.9116285240348300, 1.664130959494908, 2.198427760487026, 2.907158372862065, 3.515785969200492, 3.978110699316804, 4.909254403771076, 5.363135800573263, 5.553612119047229, 6.298927057108586, 6.828016183396789, 7.576927032515096, 7.932132252331186, 8.241314050650861, 9.079845037421382, 9.330799658151005, 10.12215404327801, 10.42129855010952, 10.86599311090632, 11.30533315216121, 12.01744548507564, 12.38158940291498, 12.99375820075330, 13.42296309840853, 13.64777632014016

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