Properties

Label 2-720-1.1-c3-0-15
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 + 34·7-s + 16·11-s + 58·13-s + 70·17-s − 4·19-s − 134·23-s + 25·25-s + 242·29-s − 100·31-s + 170·35-s − 438·37-s + 138·41-s − 178·43-s + 22·47-s + 813·49-s − 162·53-s + 80·55-s − 268·59-s + 250·61-s + 290·65-s − 422·67-s − 852·71-s + 306·73-s + 544·77-s + 456·79-s + 434·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.83·7-s + 0.438·11-s + 1.23·13-s + 0.998·17-s − 0.0482·19-s − 1.21·23-s + 1/5·25-s + 1.54·29-s − 0.579·31-s + 0.821·35-s − 1.94·37-s + 0.525·41-s − 0.631·43-s + 0.0682·47-s + 2.37·49-s − 0.419·53-s + 0.196·55-s − 0.591·59-s + 0.524·61-s + 0.553·65-s − 0.769·67-s − 1.42·71-s + 0.490·73-s + 0.805·77-s + 0.649·79-s + 0.573·83-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

\(L(2)\) \(\approx\) \(3.171541832\)
\(L(\frac12)\) \(\approx\) \(3.171541832\)
\(L(\frac{5}{2})\) 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 \( 1 - 34 T + p^{3} T^{2} \)
11 \( 1 - 16 T + p^{3} T^{2} \)
13 \( 1 - 58 T + p^{3} T^{2} \)
17 \( 1 - 70 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 + 134 T + p^{3} T^{2} \)
29 \( 1 - 242 T + p^{3} T^{2} \)
31 \( 1 + 100 T + p^{3} T^{2} \)
37 \( 1 + 438 T + p^{3} T^{2} \)
41 \( 1 - 138 T + p^{3} T^{2} \)
43 \( 1 + 178 T + p^{3} T^{2} \)
47 \( 1 - 22 T + p^{3} T^{2} \)
53 \( 1 + 162 T + p^{3} T^{2} \)
59 \( 1 + 268 T + p^{3} T^{2} \)
61 \( 1 - 250 T + p^{3} T^{2} \)
67 \( 1 + 422 T + p^{3} T^{2} \)
71 \( 1 + 12 p T + p^{3} T^{2} \)
73 \( 1 - 306 T + p^{3} T^{2} \)
79 \( 1 - 456 T + p^{3} T^{2} \)
83 \( 1 - 434 T + p^{3} T^{2} \)
89 \( 1 - 726 T + p^{3} T^{2} \)
97 \( 1 - 1378 T + p^{3} T^{2} \)
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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

−10.22572480046913745011930392958, −8.947607958223266007249640291711, −8.335377338324811771435674064641, −7.59054497170985846597906126568, −6.36464635179123227994521505831, −5.49226044370723787902006471980, −4.60805128938907848460513995844, −3.52565889202379224030494240673, −1.92193744145234397781099933324, −1.16728557410045569590268749643, 1.16728557410045569590268749643, 1.92193744145234397781099933324, 3.52565889202379224030494240673, 4.60805128938907848460513995844, 5.49226044370723787902006471980, 6.36464635179123227994521505831, 7.59054497170985846597906126568, 8.335377338324811771435674064641, 8.947607958223266007249640291711, 10.22572480046913745011930392958

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