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Properties

Label 2-87e2-1.1-c1-0-219
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  + 1.61·2-s + 0.618·4-s + 2.85·5-s + 2.23·7-s − 2.23·8-s + 4.61·10-s + 3.61·11-s + 4.23·13-s + 3.61·14-s − 4.85·16-s + 6.61·17-s + 1.85·19-s + 1.76·20-s + 5.85·22-s − 3.23·23-s + 3.14·25-s + 6.85·26-s + 1.38·28-s + 1.09·31-s − 3.38·32-s + 10.7·34-s + 6.38·35-s − 8.70·37-s + 3·38-s − 6.38·40-s + 2.85·41-s − 2.76·43-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s + 1.27·5-s + 0.845·7-s − 0.790·8-s + 1.46·10-s + 1.09·11-s + 1.17·13-s + 0.966·14-s − 1.21·16-s + 1.60·17-s + 0.425·19-s + 0.394·20-s + 1.24·22-s − 0.674·23-s + 0.629·25-s + 1.34·26-s + 0.261·28-s + 0.195·31-s − 0.597·32-s + 1.83·34-s + 1.07·35-s − 1.43·37-s + 0.486·38-s − 1.00·40-s + 0.445·41-s − 0.421·43-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 5.9225408485.922540848
L(12)L(\frac12) ≈\approx 5.9225408485.922540848
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 1−1.61T+2T2 1 - 1.61T + 2T^{2}
5 1−2.85T+5T2 1 - 2.85T + 5T^{2}
7 1−2.23T+7T2 1 - 2.23T + 7T^{2}
11 1−3.61T+11T2 1 - 3.61T + 11T^{2}
13 1−4.23T+13T2 1 - 4.23T + 13T^{2}
17 1−6.61T+17T2 1 - 6.61T + 17T^{2}
19 1−1.85T+19T2 1 - 1.85T + 19T^{2}
23 1+3.23T+23T2 1 + 3.23T + 23T^{2}
31 1−1.09T+31T2 1 - 1.09T + 31T^{2}
37 1+8.70T+37T2 1 + 8.70T + 37T^{2}
41 1−2.85T+41T2 1 - 2.85T + 41T^{2}
43 1+2.76T+43T2 1 + 2.76T + 43T^{2}
47 1−7T+47T2 1 - 7T + 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1−5.09T+59T2 1 - 5.09T + 59T^{2}
61 1−1.61T+61T2 1 - 1.61T + 61T^{2}
67 1+10.4T+67T2 1 + 10.4T + 67T^{2}
71 1+1.52T+71T2 1 + 1.52T + 71T^{2}
73 1+0.291T+73T2 1 + 0.291T + 73T^{2}
79 1−5.09T+79T2 1 - 5.09T + 79T^{2}
83 1+7.94T+83T2 1 + 7.94T + 83T^{2}
89 1−8.70T+89T2 1 - 8.70T + 89T^{2}
97 1+16.5T+97T2 1 + 16.5T + 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

−7.88459943534720628157692914467, −6.84037561779015388678618355210, −6.22673600252947180946836472461, −5.53140973067693156846699179620, −5.35783377548697195677269349012, −4.26173422144343011549242531135, −3.70633238115869278558748447001, −2.90569629734524821377489212328, −1.78482537912958157657611169535, −1.17282855779647595164503915840, 1.17282855779647595164503915840, 1.78482537912958157657611169535, 2.90569629734524821377489212328, 3.70633238115869278558748447001, 4.26173422144343011549242531135, 5.35783377548697195677269349012, 5.53140973067693156846699179620, 6.22673600252947180946836472461, 6.84037561779015388678618355210, 7.88459943534720628157692914467

Graph of the ZZ-function along the critical line