Properties

Label 2-81120-1.1-c1-0-45
Degree $2$
Conductor $81120$
Sign $-1$
Analytic cond. $647.746$
Root an. cond. $25.4508$
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  + 3-s + 5-s + 9-s + 15-s + 2·17-s − 4·19-s − 4·23-s + 25-s + 27-s − 6·29-s + 8·31-s + 6·37-s + 2·41-s − 4·43-s + 45-s − 7·49-s + 2·51-s + 6·53-s − 4·57-s − 2·61-s + 8·67-s − 4·69-s − 6·73-s + 75-s + 4·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.256·61-s + 0.977·67-s − 0.481·69-s − 0.702·73-s + 0.115·75-s + 0.450·79-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81120\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(647.746\)
Root analytic conductor: \(25.4508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{81120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 81120,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.36889030188838, −13.65565925963003, −13.29482660406583, −12.86856791243216, −12.32565592024860, −11.76624682756786, −11.25818331348163, −10.64848741660731, −10.08469652282787, −9.762074352989412, −9.261466086680797, −8.601909948785004, −8.194887524372763, −7.728870873610708, −7.112849666345933, −6.441943862994995, −6.075456167161358, −5.435318063106905, −4.767005696842333, −4.166273207819739, −3.678358053122505, −2.891829258888078, −2.386814308111070, −1.754158038611956, −1.045066424038222, 0, 1.045066424038222, 1.754158038611956, 2.386814308111070, 2.891829258888078, 3.678358053122505, 4.166273207819739, 4.767005696842333, 5.435318063106905, 6.075456167161358, 6.441943862994995, 7.112849666345933, 7.728870873610708, 8.194887524372763, 8.601909948785004, 9.261466086680797, 9.762074352989412, 10.08469652282787, 10.64848741660731, 11.25818331348163, 11.76624682756786, 12.32565592024860, 12.86856791243216, 13.29482660406583, 13.65565925963003, 14.36889030188838

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