Properties

Label 2-720-1.1-c3-0-15
Degree 22
Conductor 720720
Sign 11
Analytic cond. 42.481342.4813
Root an. cond. 6.517776.51777
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

Λ(s)=(720s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(720s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    243252^{4} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 42.481342.4813
Root analytic conductor: 6.517776.51777
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 720, ( :3/2), 1)(2,\ 720,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) \approx 3.1715418323.171541832
L(12)L(\frac12) \approx 3.1715418323.171541832
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1pT 1 - p T
good7 134T+p3T2 1 - 34 T + p^{3} T^{2}
11 116T+p3T2 1 - 16 T + p^{3} T^{2}
13 158T+p3T2 1 - 58 T + p^{3} T^{2}
17 170T+p3T2 1 - 70 T + p^{3} T^{2}
19 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
23 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
29 1242T+p3T2 1 - 242 T + p^{3} T^{2}
31 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
37 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
41 1138T+p3T2 1 - 138 T + p^{3} T^{2}
43 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
47 122T+p3T2 1 - 22 T + p^{3} T^{2}
53 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
59 1+268T+p3T2 1 + 268 T + p^{3} T^{2}
61 1250T+p3T2 1 - 250 T + p^{3} T^{2}
67 1+422T+p3T2 1 + 422 T + p^{3} T^{2}
71 1+12pT+p3T2 1 + 12 p T + p^{3} T^{2}
73 1306T+p3T2 1 - 306 T + p^{3} T^{2}
79 1456T+p3T2 1 - 456 T + p^{3} T^{2}
83 1434T+p3T2 1 - 434 T + p^{3} T^{2}
89 1726T+p3T2 1 - 726 T + p^{3} T^{2}
97 11378T+p3T2 1 - 1378 T + p^{3} T^{2}
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   L(s)=p j=12(1αj,pps)1L(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 ZZ-function along the critical line