Properties

Label 2-720-1.1-c3-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 30·7-s + 50·11-s − 20·13-s + 10·17-s + 44·19-s + 120·23-s + 25·25-s + 50·29-s − 108·31-s + 150·35-s − 40·37-s − 400·41-s − 280·43-s − 280·47-s + 557·49-s + 610·53-s + 250·55-s + 50·59-s − 518·61-s − 100·65-s + 180·67-s + 700·71-s − 410·73-s + 1.50e3·77-s + 516·79-s + 660·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.61·7-s + 1.37·11-s − 0.426·13-s + 0.142·17-s + 0.531·19-s + 1.08·23-s + 1/5·25-s + 0.320·29-s − 0.625·31-s + 0.724·35-s − 0.177·37-s − 1.52·41-s − 0.993·43-s − 0.868·47-s + 1.62·49-s + 1.58·53-s + 0.612·55-s + 0.110·59-s − 1.08·61-s − 0.190·65-s + 0.328·67-s + 1.17·71-s − 0.657·73-s + 2.22·77-s + 0.734·79-s + 0.872·83-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: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.065613366\)
\(L(\frac12)\) \(\approx\) \(3.065613366\)
\(L(\frac{5}{2})\) 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 \( 1 - 30 T + p^{3} T^{2} \)
11 \( 1 - 50 T + p^{3} T^{2} \)
13 \( 1 + 20 T + p^{3} T^{2} \)
17 \( 1 - 10 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 - 50 T + p^{3} T^{2} \)
31 \( 1 + 108 T + p^{3} T^{2} \)
37 \( 1 + 40 T + p^{3} T^{2} \)
41 \( 1 + 400 T + p^{3} T^{2} \)
43 \( 1 + 280 T + p^{3} T^{2} \)
47 \( 1 + 280 T + p^{3} T^{2} \)
53 \( 1 - 610 T + p^{3} T^{2} \)
59 \( 1 - 50 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 - 180 T + p^{3} T^{2} \)
71 \( 1 - 700 T + p^{3} T^{2} \)
73 \( 1 + 410 T + p^{3} T^{2} \)
79 \( 1 - 516 T + p^{3} T^{2} \)
83 \( 1 - 660 T + p^{3} T^{2} \)
89 \( 1 - 1500 T + p^{3} T^{2} \)
97 \( 1 + 1630 T + p^{3} T^{2} \)
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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

−9.993291932847492572798688798393, −9.074024057513124365500201026682, −8.400454716198910678421739467466, −7.38090143772431656463624801256, −6.56993996054172596977646320094, −5.31683199692416570189910112914, −4.72035791292709116070499936319, −3.48110272126592208948751840479, −1.96944074776692490227011829391, −1.12627939473566469899123802418, 1.12627939473566469899123802418, 1.96944074776692490227011829391, 3.48110272126592208948751840479, 4.72035791292709116070499936319, 5.31683199692416570189910112914, 6.56993996054172596977646320094, 7.38090143772431656463624801256, 8.400454716198910678421739467466, 9.074024057513124365500201026682, 9.993291932847492572798688798393

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