Properties

Label 2-720-1.1-c3-0-9
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 − 2·7-s − 30·11-s − 4·13-s + 90·17-s + 28·19-s − 120·23-s + 25·25-s + 210·29-s + 4·31-s − 10·35-s + 200·37-s + 240·41-s + 136·43-s + 120·47-s − 339·49-s − 30·53-s − 150·55-s + 450·59-s − 166·61-s − 20·65-s − 908·67-s + 1.02e3·71-s − 250·73-s + 60·77-s + 916·79-s + 1.14e3·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.107·7-s − 0.822·11-s − 0.0853·13-s + 1.28·17-s + 0.338·19-s − 1.08·23-s + 1/5·25-s + 1.34·29-s + 0.0231·31-s − 0.0482·35-s + 0.888·37-s + 0.914·41-s + 0.482·43-s + 0.372·47-s − 0.988·49-s − 0.0777·53-s − 0.367·55-s + 0.992·59-s − 0.348·61-s − 0.0381·65-s − 1.65·67-s + 1.70·71-s − 0.400·73-s + 0.0888·77-s + 1.30·79-s + 1.50·83-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

\(L(2)\) \(\approx\) \(2.097660147\)
\(L(\frac12)\) \(\approx\) \(2.097660147\)
\(L(\frac{5}{2})\) 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 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 30 T + p^{3} T^{2} \)
13 \( 1 + 4 T + p^{3} T^{2} \)
17 \( 1 - 90 T + p^{3} T^{2} \)
19 \( 1 - 28 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 - 210 T + p^{3} T^{2} \)
31 \( 1 - 4 T + p^{3} T^{2} \)
37 \( 1 - 200 T + p^{3} T^{2} \)
41 \( 1 - 240 T + p^{3} T^{2} \)
43 \( 1 - 136 T + p^{3} T^{2} \)
47 \( 1 - 120 T + p^{3} T^{2} \)
53 \( 1 + 30 T + p^{3} T^{2} \)
59 \( 1 - 450 T + p^{3} T^{2} \)
61 \( 1 + 166 T + p^{3} T^{2} \)
67 \( 1 + 908 T + p^{3} T^{2} \)
71 \( 1 - 1020 T + p^{3} T^{2} \)
73 \( 1 + 250 T + p^{3} T^{2} \)
79 \( 1 - 916 T + p^{3} T^{2} \)
83 \( 1 - 1140 T + p^{3} T^{2} \)
89 \( 1 + 420 T + p^{3} T^{2} \)
97 \( 1 - 1538 T + p^{3} T^{2} \)
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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

−10.04471924250742576953908626384, −9.317508592337488792506012694306, −8.141219661259370071311111900127, −7.57541439135795731991903397840, −6.33865021461451326618461562541, −5.59286393147008988509436061423, −4.62320870206117857577710914036, −3.31655207623427489151424090499, −2.28504536800406333302608894255, −0.837824226998049038090069762986, 0.837824226998049038090069762986, 2.28504536800406333302608894255, 3.31655207623427489151424090499, 4.62320870206117857577710914036, 5.59286393147008988509436061423, 6.33865021461451326618461562541, 7.57541439135795731991903397840, 8.141219661259370071311111900127, 9.317508592337488792506012694306, 10.04471924250742576953908626384

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