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Properties

Label 2-87e2-1.1-c1-0-22
Degree 22
Conductor 75697569
Sign 11
Analytic cond. 60.438760.4387
Root an. cond. 7.774237.77423
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 1.61·4-s − 3.85·5-s − 2.23·7-s + 2.23·8-s + 2.38·10-s + 1.38·11-s − 0.236·13-s + 1.38·14-s + 1.85·16-s + 4.38·17-s − 4.85·19-s + 6.23·20-s − 0.854·22-s + 1.23·23-s + 9.85·25-s + 0.145·26-s + 3.61·28-s − 10.0·31-s − 5.61·32-s − 2.70·34-s + 8.61·35-s + 4.70·37-s + 3.00·38-s − 8.61·40-s − 3.85·41-s − 7.23·43-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s − 1.72·5-s − 0.845·7-s + 0.790·8-s + 0.753·10-s + 0.416·11-s − 0.0654·13-s + 0.369·14-s + 0.463·16-s + 1.06·17-s − 1.11·19-s + 1.39·20-s − 0.182·22-s + 0.257·23-s + 1.97·25-s + 0.0286·26-s + 0.683·28-s − 1.81·31-s − 0.993·32-s − 0.464·34-s + 1.45·35-s + 0.774·37-s + 0.486·38-s − 1.36·40-s − 0.601·41-s − 1.10·43-s + ⋯

Functional equation

Λ(s)=(7569s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7569s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 75697569    =    32⋅2923^{2} \cdot 29^{2}
Sign: 11
Analytic conductor: 60.438760.4387
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7569, ( :1/2), 1)(2,\ 7569,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.27096230260.2709623026
L(12)L(\frac12) ≈\approx 0.27096230260.2709623026
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
29 1 1
good2 1+0.618T+2T2 1 + 0.618T + 2T^{2}
5 1+3.85T+5T2 1 + 3.85T + 5T^{2}
7 1+2.23T+7T2 1 + 2.23T + 7T^{2}
11 1−1.38T+11T2 1 - 1.38T + 11T^{2}
13 1+0.236T+13T2 1 + 0.236T + 13T^{2}
17 1−4.38T+17T2 1 - 4.38T + 17T^{2}
19 1+4.85T+19T2 1 + 4.85T + 19T^{2}
23 1−1.23T+23T2 1 - 1.23T + 23T^{2}
31 1+10.0T+31T2 1 + 10.0T + 31T^{2}
37 1−4.70T+37T2 1 - 4.70T + 37T^{2}
41 1+3.85T+41T2 1 + 3.85T + 41T^{2}
43 1+7.23T+43T2 1 + 7.23T + 43T^{2}
47 1−7T+47T2 1 - 7T + 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1+6.09T+59T2 1 + 6.09T + 59T^{2}
61 1+0.618T+61T2 1 + 0.618T + 61T^{2}
67 1+1.52T+67T2 1 + 1.52T + 67T^{2}
71 1+10.4T+71T2 1 + 10.4T + 71T^{2}
73 1+13.7T+73T2 1 + 13.7T + 73T^{2}
79 1+6.09T+79T2 1 + 6.09T + 79T^{2}
83 1−9.94T+83T2 1 - 9.94T + 83T^{2}
89 1+4.70T+89T2 1 + 4.70T + 89T^{2}
97 1−3.56T+97T2 1 - 3.56T + 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

−7.83537674822379891727120988647, −7.44017077797015206223541996258, −6.72017767568210285323956542016, −5.78410466072182894272959827141, −4.88793211929119153623588473287, −4.13267674993182048909767688163, −3.68208429220805701304412300669, −3.00317137124081724324657105516, −1.43571576194055332557475879467, −0.29756994223083770618652804036, 0.29756994223083770618652804036, 1.43571576194055332557475879467, 3.00317137124081724324657105516, 3.68208429220805701304412300669, 4.13267674993182048909767688163, 4.88793211929119153623588473287, 5.78410466072182894272959827141, 6.72017767568210285323956542016, 7.44017077797015206223541996258, 7.83537674822379891727120988647

Graph of the ZZ-function along the critical line