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Properties

Label 2-8001-1.1-c1-0-247
Degree 22
Conductor 80018001
Sign −1-1
Analytic cond. 63.888363.8883
Root an. cond. 7.993017.99301
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.691·2-s − 1.52·4-s + 0.236·5-s + 7-s − 2.43·8-s + 0.163·10-s + 4.73·11-s − 1.51·13-s + 0.691·14-s + 1.35·16-s − 4.88·17-s + 2.13·19-s − 0.360·20-s + 3.27·22-s − 1.62·23-s − 4.94·25-s − 1.05·26-s − 1.52·28-s + 0.481·29-s − 9.10·31-s + 5.81·32-s − 3.37·34-s + 0.236·35-s + 10.9·37-s + 1.47·38-s − 0.576·40-s + 4.78·41-s + ⋯
L(s)  = 1  + 0.489·2-s − 0.760·4-s + 0.105·5-s + 0.377·7-s − 0.861·8-s + 0.0517·10-s + 1.42·11-s − 0.421·13-s + 0.184·14-s + 0.339·16-s − 1.18·17-s + 0.490·19-s − 0.0805·20-s + 0.697·22-s − 0.337·23-s − 0.988·25-s − 0.206·26-s − 0.287·28-s + 0.0894·29-s − 1.63·31-s + 1.02·32-s − 0.579·34-s + 0.0400·35-s + 1.80·37-s + 0.239·38-s − 0.0911·40-s + 0.746·41-s + ⋯

Functional equation

Λ(s)=(8001s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8001s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 80018001    =    32⋅7⋅1273^{2} \cdot 7 \cdot 127
Sign: −1-1
Analytic conductor: 63.888363.8883
Root analytic conductor: 7.993017.99301
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8001, ( :1/2), −1)(2,\ 8001,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad3 1 1
7 1−T 1 - T
127 1−T 1 - T
good2 1−0.691T+2T2 1 - 0.691T + 2T^{2}
5 1−0.236T+5T2 1 - 0.236T + 5T^{2}
11 1−4.73T+11T2 1 - 4.73T + 11T^{2}
13 1+1.51T+13T2 1 + 1.51T + 13T^{2}
17 1+4.88T+17T2 1 + 4.88T + 17T^{2}
19 1−2.13T+19T2 1 - 2.13T + 19T^{2}
23 1+1.62T+23T2 1 + 1.62T + 23T^{2}
29 1−0.481T+29T2 1 - 0.481T + 29T^{2}
31 1+9.10T+31T2 1 + 9.10T + 31T^{2}
37 1−10.9T+37T2 1 - 10.9T + 37T^{2}
41 1−4.78T+41T2 1 - 4.78T + 41T^{2}
43 1−1.26T+43T2 1 - 1.26T + 43T^{2}
47 1+6.43T+47T2 1 + 6.43T + 47T^{2}
53 1+11.2T+53T2 1 + 11.2T + 53T^{2}
59 1−7.68T+59T2 1 - 7.68T + 59T^{2}
61 1−5.26T+61T2 1 - 5.26T + 61T^{2}
67 1+15.1T+67T2 1 + 15.1T + 67T^{2}
71 1+2.66T+71T2 1 + 2.66T + 71T^{2}
73 1−10.5T+73T2 1 - 10.5T + 73T^{2}
79 1+13.1T+79T2 1 + 13.1T + 79T^{2}
83 1−11.1T+83T2 1 - 11.1T + 83T^{2}
89 1+11.0T+89T2 1 + 11.0T + 89T^{2}
97 1+9.66T+97T2 1 + 9.66T + 97T^{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

−7.50307010205427591832263403108, −6.65061649402412208465009787998, −6.00305982061410108299613073708, −5.37950974133863637072081978063, −4.42643415634495477655977330659, −4.16311630772601057797788855158, −3.32167762417548066963593158872, −2.26393676121810750621959709920, −1.28736542700736629530522306373, 0, 1.28736542700736629530522306373, 2.26393676121810750621959709920, 3.32167762417548066963593158872, 4.16311630772601057797788855158, 4.42643415634495477655977330659, 5.37950974133863637072081978063, 6.00305982061410108299613073708, 6.65061649402412208465009787998, 7.50307010205427591832263403108

Graph of the ZZ-function along the critical line