Properties

Label 2-720-1.1-c3-0-6
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 + 16·7-s − 60·11-s + 86·13-s − 18·17-s − 44·19-s + 48·23-s + 25·25-s + 186·29-s − 176·31-s − 80·35-s + 254·37-s − 186·41-s + 100·43-s + 168·47-s − 87·49-s + 498·53-s + 300·55-s − 252·59-s − 58·61-s − 430·65-s + 1.03e3·67-s + 168·71-s + 506·73-s − 960·77-s − 272·79-s + 948·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.863·7-s − 1.64·11-s + 1.83·13-s − 0.256·17-s − 0.531·19-s + 0.435·23-s + 1/5·25-s + 1.19·29-s − 1.01·31-s − 0.386·35-s + 1.12·37-s − 0.708·41-s + 0.354·43-s + 0.521·47-s − 0.253·49-s + 1.29·53-s + 0.735·55-s − 0.556·59-s − 0.121·61-s − 0.820·65-s + 1.88·67-s + 0.280·71-s + 0.811·73-s − 1.42·77-s − 0.387·79-s + 1.25·83-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

\(L(2)\) \(\approx\) \(1.926327851\)
\(L(\frac12)\) \(\approx\) \(1.926327851\)
\(L(\frac{5}{2})\) 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 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 + 60 T + p^{3} T^{2} \)
13 \( 1 - 86 T + p^{3} T^{2} \)
17 \( 1 + 18 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 - 48 T + p^{3} T^{2} \)
29 \( 1 - 186 T + p^{3} T^{2} \)
31 \( 1 + 176 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 + 186 T + p^{3} T^{2} \)
43 \( 1 - 100 T + p^{3} T^{2} \)
47 \( 1 - 168 T + p^{3} T^{2} \)
53 \( 1 - 498 T + p^{3} T^{2} \)
59 \( 1 + 252 T + p^{3} T^{2} \)
61 \( 1 + 58 T + p^{3} T^{2} \)
67 \( 1 - 1036 T + p^{3} T^{2} \)
71 \( 1 - 168 T + p^{3} T^{2} \)
73 \( 1 - 506 T + p^{3} T^{2} \)
79 \( 1 + 272 T + p^{3} T^{2} \)
83 \( 1 - 948 T + p^{3} T^{2} \)
89 \( 1 - 1014 T + p^{3} T^{2} \)
97 \( 1 + 766 T + p^{3} T^{2} \)
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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

−10.27745100394038868537388559845, −8.878424474206252977828057078114, −8.270097465629327927083415193999, −7.63600882673980566928377164255, −6.46187550071439174414239950958, −5.44102095769878366139880357543, −4.56498944508401861710770170494, −3.47420223056340932109716453053, −2.21806463317944159974550574708, −0.805483469972414940921634019192, 0.805483469972414940921634019192, 2.21806463317944159974550574708, 3.47420223056340932109716453053, 4.56498944508401861710770170494, 5.44102095769878366139880357543, 6.46187550071439174414239950958, 7.63600882673980566928377164255, 8.270097465629327927083415193999, 8.878424474206252977828057078114, 10.27745100394038868537388559845

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