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Properties

Label 2-85-1.1-c1-0-0
Degree 22
Conductor 8585
Sign 11
Analytic cond. 0.6787280.678728
Root an. cond. 0.8238490.823849
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  − 1.73·2-s + 2.73·3-s + 0.999·4-s + 5-s − 4.73·6-s − 2.73·7-s + 1.73·8-s + 4.46·9-s − 1.73·10-s + 4.73·11-s + 2.73·12-s − 4·13-s + 4.73·14-s + 2.73·15-s − 5·16-s − 17-s − 7.73·18-s − 1.46·19-s + 0.999·20-s − 7.46·21-s − 8.19·22-s − 8.19·23-s + 4.73·24-s + 25-s + 6.92·26-s + 3.99·27-s − 2.73·28-s + ⋯
L(s)  = 1  − 1.22·2-s + 1.57·3-s + 0.499·4-s + 0.447·5-s − 1.93·6-s − 1.03·7-s + 0.612·8-s + 1.48·9-s − 0.547·10-s + 1.42·11-s + 0.788·12-s − 1.10·13-s + 1.26·14-s + 0.705·15-s − 1.25·16-s − 0.242·17-s − 1.82·18-s − 0.335·19-s + 0.223·20-s − 1.62·21-s − 1.74·22-s − 1.70·23-s + 0.965·24-s + 0.200·25-s + 1.35·26-s + 0.769·27-s − 0.516·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 11
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 1)(2,\ 85,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.82139186900.8213918690
L(12)L(\frac12) ≈\approx 0.82139186900.8213918690
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)
bad5 1−T 1 - T
17 1+T 1 + T
good2 1+1.73T+2T2 1 + 1.73T + 2T^{2}
3 1−2.73T+3T2 1 - 2.73T + 3T^{2}
7 1+2.73T+7T2 1 + 2.73T + 7T^{2}
11 1−4.73T+11T2 1 - 4.73T + 11T^{2}
13 1+4T+13T2 1 + 4T + 13T^{2}
19 1+1.46T+19T2 1 + 1.46T + 19T^{2}
23 1+8.19T+23T2 1 + 8.19T + 23T^{2}
29 1+3.46T+29T2 1 + 3.46T + 29T^{2}
31 1−3.26T+31T2 1 - 3.26T + 31T^{2}
37 1+0.535T+37T2 1 + 0.535T + 37T^{2}
41 1+3.46T+41T2 1 + 3.46T + 41T^{2}
43 1+0.535T+43T2 1 + 0.535T + 43T^{2}
47 1−12.9T+47T2 1 - 12.9T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−2.53T+59T2 1 - 2.53T + 59T^{2}
61 1+4.92T+61T2 1 + 4.92T + 61T^{2}
67 1+10T+67T2 1 + 10T + 67T^{2}
71 1−11.6T+71T2 1 - 11.6T + 71T^{2}
73 1−6.39T+73T2 1 - 6.39T + 73T^{2}
79 1−14.5T+79T2 1 - 14.5T + 79T^{2}
83 1−8.53T+83T2 1 - 8.53T + 83T^{2}
89 1−4.39T+89T2 1 - 4.39T + 89T^{2}
97 1+4.92T+97T2 1 + 4.92T + 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

−14.18010413167243495194910902705, −13.50753370512687398199216081828, −12.18240140832389069041995272520, −10.22908803495156393026872439186, −9.492184568279700076490858135884, −8.969589372484756977018515056691, −7.79032767204837674777103178108, −6.63915818226960438386848743997, −3.95656870881855527374068847494, −2.17582772087516134866370577695, 2.17582772087516134866370577695, 3.95656870881855527374068847494, 6.63915818226960438386848743997, 7.79032767204837674777103178108, 8.969589372484756977018515056691, 9.492184568279700076490858135884, 10.22908803495156393026872439186, 12.18240140832389069041995272520, 13.50753370512687398199216081828, 14.18010413167243495194910902705

Graph of the ZZ-function along the critical line