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Properties

Label 2-1440-1.1-c3-0-45
Degree 22
Conductor 14401440
Sign −1-1
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 16·7-s − 24·11-s − 14·13-s + 18·17-s + 36·19-s − 104·23-s + 25·25-s + 250·29-s − 28·31-s − 80·35-s − 54·37-s − 354·41-s + 228·43-s − 408·47-s − 87·49-s − 262·53-s + 120·55-s + 64·59-s + 374·61-s + 70·65-s + 300·67-s − 1.01e3·71-s + 274·73-s − 384·77-s + 788·79-s + 396·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.863·7-s − 0.657·11-s − 0.298·13-s + 0.256·17-s + 0.434·19-s − 0.942·23-s + 1/5·25-s + 1.60·29-s − 0.162·31-s − 0.386·35-s − 0.239·37-s − 1.34·41-s + 0.808·43-s − 1.26·47-s − 0.253·49-s − 0.679·53-s + 0.294·55-s + 0.141·59-s + 0.785·61-s + 0.133·65-s + 0.547·67-s − 1.69·71-s + 0.439·73-s − 0.568·77-s + 1.12·79-s + 0.523·83-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1440, ( :3/2), −1)(2,\ 1440,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1+pT 1 + p T
good7 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
11 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
13 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
17 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
19 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
23 1+104T+p3T2 1 + 104 T + p^{3} T^{2}
29 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
31 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
37 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
41 1+354T+p3T2 1 + 354 T + p^{3} T^{2}
43 1−228T+p3T2 1 - 228 T + p^{3} T^{2}
47 1+408T+p3T2 1 + 408 T + p^{3} T^{2}
53 1+262T+p3T2 1 + 262 T + p^{3} T^{2}
59 1−64T+p3T2 1 - 64 T + p^{3} T^{2}
61 1−374T+p3T2 1 - 374 T + p^{3} T^{2}
67 1−300T+p3T2 1 - 300 T + p^{3} T^{2}
71 1+1016T+p3T2 1 + 1016 T + p^{3} T^{2}
73 1−274T+p3T2 1 - 274 T + p^{3} T^{2}
79 1−788T+p3T2 1 - 788 T + p^{3} T^{2}
83 1−396T+p3T2 1 - 396 T + p^{3} T^{2}
89 1+786T+p3T2 1 + 786 T + p^{3} T^{2}
97 1+1086T+p3T2 1 + 1086 T + p^{3} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−8.467329379983277083320153500468, −8.069973264472887356272052356758, −7.29392201834200258595744675797, −6.32941905717651427389945483809, −5.22175418206935184668768722913, −4.66991757526515338218738024816, −3.56719507961430006996569248390, −2.50753651099500834346477758792, −1.33999268779207022759395489012, 0, 1.33999268779207022759395489012, 2.50753651099500834346477758792, 3.56719507961430006996569248390, 4.66991757526515338218738024816, 5.22175418206935184668768722913, 6.32941905717651427389945483809, 7.29392201834200258595744675797, 8.069973264472887356272052356758, 8.467329379983277083320153500468

Graph of the ZZ-function along the critical line