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Properties

Label 2-85-1.1-c3-0-0
Degree 22
Conductor 8585
Sign 11
Analytic cond. 5.015165.01516
Root an. cond. 2.239452.23945
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.64·2-s − 5.37·3-s − 5.31·4-s − 5·5-s + 8.81·6-s + 2.20·7-s + 21.8·8-s + 1.88·9-s + 8.20·10-s + 18.7·11-s + 28.5·12-s + 62.9·13-s − 3.60·14-s + 26.8·15-s + 6.68·16-s − 17·17-s − 3.09·18-s − 47.8·19-s + 26.5·20-s − 11.8·21-s − 30.7·22-s + 153.·23-s − 117.·24-s + 25·25-s − 103.·26-s + 134.·27-s − 11.6·28-s + ⋯
L(s)  = 1  − 0.579·2-s − 1.03·3-s − 0.663·4-s − 0.447·5-s + 0.599·6-s + 0.118·7-s + 0.964·8-s + 0.0698·9-s + 0.259·10-s + 0.514·11-s + 0.686·12-s + 1.34·13-s − 0.0689·14-s + 0.462·15-s + 0.104·16-s − 0.242·17-s − 0.0405·18-s − 0.577·19-s + 0.296·20-s − 0.122·21-s − 0.298·22-s + 1.39·23-s − 0.997·24-s + 0.200·25-s − 0.778·26-s + 0.962·27-s − 0.0788·28-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 11
Analytic conductor: 5.015165.01516
Root analytic conductor: 2.239452.23945
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 85, ( :3/2), 1)(2,\ 85,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.58506027260.5850602726
L(12)L(\frac12) ≈\approx 0.58506027260.5850602726
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)
bad5 1+5T 1 + 5T
17 1+17T 1 + 17T
good2 1+1.64T+8T2 1 + 1.64T + 8T^{2}
3 1+5.37T+27T2 1 + 5.37T + 27T^{2}
7 1−2.20T+343T2 1 - 2.20T + 343T^{2}
11 1−18.7T+1.33e3T2 1 - 18.7T + 1.33e3T^{2}
13 1−62.9T+2.19e3T2 1 - 62.9T + 2.19e3T^{2}
19 1+47.8T+6.85e3T2 1 + 47.8T + 6.85e3T^{2}
23 1−153.T+1.21e4T2 1 - 153.T + 1.21e4T^{2}
29 1+64.4T+2.43e4T2 1 + 64.4T + 2.43e4T^{2}
31 1+40.9T+2.97e4T2 1 + 40.9T + 2.97e4T^{2}
37 1−32.6T+5.06e4T2 1 - 32.6T + 5.06e4T^{2}
41 1−159.T+6.89e4T2 1 - 159.T + 6.89e4T^{2}
43 1+111.T+7.95e4T2 1 + 111.T + 7.95e4T^{2}
47 1−614.T+1.03e5T2 1 - 614.T + 1.03e5T^{2}
53 1−308.T+1.48e5T2 1 - 308.T + 1.48e5T^{2}
59 1−267.T+2.05e5T2 1 - 267.T + 2.05e5T^{2}
61 1−521.T+2.26e5T2 1 - 521.T + 2.26e5T^{2}
67 1−118.T+3.00e5T2 1 - 118.T + 3.00e5T^{2}
71 1−1.14e3T+3.57e5T2 1 - 1.14e3T + 3.57e5T^{2}
73 1+40.7T+3.89e5T2 1 + 40.7T + 3.89e5T^{2}
79 1−374.T+4.93e5T2 1 - 374.T + 4.93e5T^{2}
83 1−826.T+5.71e5T2 1 - 826.T + 5.71e5T^{2}
89 1+38.9T+7.04e5T2 1 + 38.9T + 7.04e5T^{2}
97 1+917.T+9.12e5T2 1 + 917.T + 9.12e5T^{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

−13.62064953262927678806666962454, −12.57742948317407991377594083788, −11.26615596369562402083065683204, −10.72665792889003534743245662327, −9.179278558063589002750104557368, −8.300035359640679909893669673492, −6.75505135012596944977442189398, −5.37220924724512324166708663936, −4.00385662350735805975507730683, −0.844794642058816184049789383884, 0.844794642058816184049789383884, 4.00385662350735805975507730683, 5.37220924724512324166708663936, 6.75505135012596944977442189398, 8.300035359640679909893669673492, 9.179278558063589002750104557368, 10.72665792889003534743245662327, 11.26615596369562402083065683204, 12.57742948317407991377594083788, 13.62064953262927678806666962454

Graph of the ZZ-function along the critical line