Properties

Label 2-720-1.1-c3-0-11
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 − 4·7-s + 72·11-s − 6·13-s − 38·17-s − 52·19-s + 152·23-s + 25·25-s + 78·29-s − 120·31-s − 20·35-s − 150·37-s − 362·41-s + 484·43-s + 280·47-s − 327·49-s + 670·53-s + 360·55-s + 696·59-s + 222·61-s − 30·65-s + 4·67-s + 96·71-s + 178·73-s − 288·77-s + 632·79-s − 612·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.215·7-s + 1.97·11-s − 0.128·13-s − 0.542·17-s − 0.627·19-s + 1.37·23-s + 1/5·25-s + 0.499·29-s − 0.695·31-s − 0.0965·35-s − 0.666·37-s − 1.37·41-s + 1.71·43-s + 0.868·47-s − 0.953·49-s + 1.73·53-s + 0.882·55-s + 1.53·59-s + 0.465·61-s − 0.0572·65-s + 0.00729·67-s + 0.160·71-s + 0.285·73-s − 0.426·77-s + 0.900·79-s − 0.809·83-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

\(L(2)\) \(\approx\) \(2.424086190\)
\(L(\frac12)\) \(\approx\) \(2.424086190\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 - 72 T + p^{3} T^{2} \)
13 \( 1 + 6 T + p^{3} T^{2} \)
17 \( 1 + 38 T + p^{3} T^{2} \)
19 \( 1 + 52 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 - 78 T + p^{3} T^{2} \)
31 \( 1 + 120 T + p^{3} T^{2} \)
37 \( 1 + 150 T + p^{3} T^{2} \)
41 \( 1 + 362 T + p^{3} T^{2} \)
43 \( 1 - 484 T + p^{3} T^{2} \)
47 \( 1 - 280 T + p^{3} T^{2} \)
53 \( 1 - 670 T + p^{3} T^{2} \)
59 \( 1 - 696 T + p^{3} T^{2} \)
61 \( 1 - 222 T + p^{3} T^{2} \)
67 \( 1 - 4 T + p^{3} T^{2} \)
71 \( 1 - 96 T + p^{3} T^{2} \)
73 \( 1 - 178 T + p^{3} T^{2} \)
79 \( 1 - 8 p T + p^{3} T^{2} \)
83 \( 1 + 612 T + p^{3} T^{2} \)
89 \( 1 + 994 T + p^{3} T^{2} \)
97 \( 1 - 1634 T + p^{3} T^{2} \)
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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

−9.927879345687842411253188442127, −9.040359145508973512920696820811, −8.655331802412648001115257485215, −7.06287814534809035049817351253, −6.64789042668547664881110348912, −5.61173594436924589927459403878, −4.44214354041109251000216738143, −3.51235449494208983025107828017, −2.13660290910793640930210806454, −0.933719639472869479412692177614, 0.933719639472869479412692177614, 2.13660290910793640930210806454, 3.51235449494208983025107828017, 4.44214354041109251000216738143, 5.61173594436924589927459403878, 6.64789042668547664881110348912, 7.06287814534809035049817351253, 8.655331802412648001115257485215, 9.040359145508973512920696820811, 9.927879345687842411253188442127

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