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Properties

Label 2-8512-1.1-c1-0-130
Degree 22
Conductor 85128512
Sign 11
Analytic cond. 67.968667.9686
Root an. cond. 8.244318.24431
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  + 3.19·3-s − 1.16·5-s + 7-s + 7.19·9-s + 5.39·11-s + 1.73·13-s − 3.72·15-s − 1.79·17-s + 19-s + 3.19·21-s − 2.08·23-s − 3.63·25-s + 13.3·27-s − 3.75·29-s + 3.79·31-s + 17.2·33-s − 1.16·35-s + 1.78·37-s + 5.53·39-s + 8.34·41-s − 2.48·43-s − 8.39·45-s + 5.30·47-s + 49-s − 5.71·51-s + 0.355·53-s − 6.30·55-s + ⋯
L(s)  = 1  + 1.84·3-s − 0.522·5-s + 0.377·7-s + 2.39·9-s + 1.62·11-s + 0.480·13-s − 0.962·15-s − 0.434·17-s + 0.229·19-s + 0.696·21-s − 0.435·23-s − 0.727·25-s + 2.57·27-s − 0.696·29-s + 0.680·31-s + 3.00·33-s − 0.197·35-s + 0.292·37-s + 0.885·39-s + 1.30·41-s − 0.379·43-s − 1.25·45-s + 0.774·47-s + 0.142·49-s − 0.800·51-s + 0.0488·53-s − 0.849·55-s + ⋯

Functional equation

Λ(s)=(8512s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8512s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85128512    =    26⋅7⋅192^{6} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 67.968667.9686
Root analytic conductor: 8.244318.24431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8512, ( :1/2), 1)(2,\ 8512,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.9472610114.947261011
L(12)L(\frac12) ≈\approx 4.9472610114.947261011
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)
bad2 1 1
7 1−T 1 - T
19 1−T 1 - T
good3 1−3.19T+3T2 1 - 3.19T + 3T^{2}
5 1+1.16T+5T2 1 + 1.16T + 5T^{2}
11 1−5.39T+11T2 1 - 5.39T + 11T^{2}
13 1−1.73T+13T2 1 - 1.73T + 13T^{2}
17 1+1.79T+17T2 1 + 1.79T + 17T^{2}
23 1+2.08T+23T2 1 + 2.08T + 23T^{2}
29 1+3.75T+29T2 1 + 3.75T + 29T^{2}
31 1−3.79T+31T2 1 - 3.79T + 31T^{2}
37 1−1.78T+37T2 1 - 1.78T + 37T^{2}
41 1−8.34T+41T2 1 - 8.34T + 41T^{2}
43 1+2.48T+43T2 1 + 2.48T + 43T^{2}
47 1−5.30T+47T2 1 - 5.30T + 47T^{2}
53 1−0.355T+53T2 1 - 0.355T + 53T^{2}
59 1−12.9T+59T2 1 - 12.9T + 59T^{2}
61 1+5.52T+61T2 1 + 5.52T + 61T^{2}
67 1−7.98T+67T2 1 - 7.98T + 67T^{2}
71 1+9.55T+71T2 1 + 9.55T + 71T^{2}
73 1+6.65T+73T2 1 + 6.65T + 73T^{2}
79 1−5.39T+79T2 1 - 5.39T + 79T^{2}
83 1+1.80T+83T2 1 + 1.80T + 83T^{2}
89 1+7.13T+89T2 1 + 7.13T + 89T^{2}
97 1+15.2T+97T2 1 + 15.2T + 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

−7.83194338444419474201704120677, −7.35605443070910934219327434808, −6.65372417449940990494826517754, −5.83584740499722414748750476563, −4.53192887989612724201959672065, −3.98606358511433435601850117274, −3.64945430738242362554342540131, −2.67033283258827167698856788834, −1.87263005678221448891748804803, −1.08475015464377075407662070246, 1.08475015464377075407662070246, 1.87263005678221448891748804803, 2.67033283258827167698856788834, 3.64945430738242362554342540131, 3.98606358511433435601850117274, 4.53192887989612724201959672065, 5.83584740499722414748750476563, 6.65372417449940990494826517754, 7.35605443070910934219327434808, 7.83194338444419474201704120677

Graph of the ZZ-function along the critical line