Properties

Label 2-720-1.1-c3-0-17
Degree $2$
Conductor $720$
Sign $-1$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 14·7-s − 6·11-s + 68·13-s + 78·17-s − 44·19-s − 120·23-s + 25·25-s + 126·29-s + 244·31-s + 70·35-s − 304·37-s − 480·41-s − 104·43-s − 600·47-s − 147·49-s − 258·53-s + 30·55-s − 534·59-s + 362·61-s − 340·65-s + 268·67-s + 972·71-s + 470·73-s + 84·77-s − 1.24e3·79-s − 396·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.164·11-s + 1.45·13-s + 1.11·17-s − 0.531·19-s − 1.08·23-s + 1/5·25-s + 0.806·29-s + 1.41·31-s + 0.338·35-s − 1.35·37-s − 1.82·41-s − 0.368·43-s − 1.86·47-s − 3/7·49-s − 0.668·53-s + 0.0735·55-s − 1.17·59-s + 0.759·61-s − 0.648·65-s + 0.488·67-s + 1.62·71-s + 0.753·73-s + 0.124·77-s − 1.77·79-s − 0.523·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good7 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 + 6 T + p^{3} T^{2} \)
13 \( 1 - 68 T + p^{3} T^{2} \)
17 \( 1 - 78 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 - 126 T + p^{3} T^{2} \)
31 \( 1 - 244 T + p^{3} T^{2} \)
37 \( 1 + 304 T + p^{3} T^{2} \)
41 \( 1 + 480 T + p^{3} T^{2} \)
43 \( 1 + 104 T + p^{3} T^{2} \)
47 \( 1 + 600 T + p^{3} T^{2} \)
53 \( 1 + 258 T + p^{3} T^{2} \)
59 \( 1 + 534 T + p^{3} T^{2} \)
61 \( 1 - 362 T + p^{3} T^{2} \)
67 \( 1 - 4 p T + p^{3} T^{2} \)
71 \( 1 - 972 T + p^{3} T^{2} \)
73 \( 1 - 470 T + p^{3} T^{2} \)
79 \( 1 + 1244 T + p^{3} T^{2} \)
83 \( 1 + 396 T + p^{3} T^{2} \)
89 \( 1 + 972 T + p^{3} T^{2} \)
97 \( 1 + 46 T + p^{3} T^{2} \)
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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

−9.792353281759657636053981582480, −8.421310134144132243852317106789, −8.168906571301881116353776002870, −6.74881896626155729834288495618, −6.20363913881512636114174349153, −5.02781413603319067801693076954, −3.78791583437946911555016635701, −3.10608834109513979569047647138, −1.44858661153723415934145873834, 0, 1.44858661153723415934145873834, 3.10608834109513979569047647138, 3.78791583437946911555016635701, 5.02781413603319067801693076954, 6.20363913881512636114174349153, 6.74881896626155729834288495618, 8.168906571301881116353776002870, 8.421310134144132243852317106789, 9.792353281759657636053981582480

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