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Properties

Label 2-8512-1.1-c1-0-131
Degree 22
Conductor 85128512
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.185·3-s − 0.737·5-s − 7-s − 2.96·9-s − 0.814·11-s − 0.677·13-s + 0.137·15-s + 3.00·17-s + 19-s + 0.185·21-s + 3.24·23-s − 4.45·25-s + 1.10·27-s + 7.57·29-s − 0.853·31-s + 0.151·33-s + 0.737·35-s − 0.870·37-s + 0.125·39-s + 1.42·41-s − 1.10·43-s + 2.18·45-s − 1.87·47-s + 49-s − 0.559·51-s + 5.00·53-s + 0.600·55-s + ⋯
L(s)  = 1  − 0.107·3-s − 0.329·5-s − 0.377·7-s − 0.988·9-s − 0.245·11-s − 0.187·13-s + 0.0353·15-s + 0.729·17-s + 0.229·19-s + 0.0405·21-s + 0.676·23-s − 0.891·25-s + 0.213·27-s + 1.40·29-s − 0.153·31-s + 0.0263·33-s + 0.124·35-s − 0.143·37-s + 0.0201·39-s + 0.223·41-s − 0.168·43-s + 0.325·45-s − 0.273·47-s + 0.142·49-s − 0.0782·51-s + 0.687·53-s + 0.0809·55-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: −1-1
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: 11
Selberg data: (2, 8512, ( :1/2), −1)(2,\ 8512,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+0.185T+3T2 1 + 0.185T + 3T^{2}
5 1+0.737T+5T2 1 + 0.737T + 5T^{2}
11 1+0.814T+11T2 1 + 0.814T + 11T^{2}
13 1+0.677T+13T2 1 + 0.677T + 13T^{2}
17 1−3.00T+17T2 1 - 3.00T + 17T^{2}
23 1−3.24T+23T2 1 - 3.24T + 23T^{2}
29 1−7.57T+29T2 1 - 7.57T + 29T^{2}
31 1+0.853T+31T2 1 + 0.853T + 31T^{2}
37 1+0.870T+37T2 1 + 0.870T + 37T^{2}
41 1−1.42T+41T2 1 - 1.42T + 41T^{2}
43 1+1.10T+43T2 1 + 1.10T + 43T^{2}
47 1+1.87T+47T2 1 + 1.87T + 47T^{2}
53 1−5.00T+53T2 1 - 5.00T + 53T^{2}
59 1−6.54T+59T2 1 - 6.54T + 59T^{2}
61 1−2.00T+61T2 1 - 2.00T + 61T^{2}
67 1+12.4T+67T2 1 + 12.4T + 67T^{2}
71 1−8.86T+71T2 1 - 8.86T + 71T^{2}
73 1+12.6T+73T2 1 + 12.6T + 73T^{2}
79 1+1.72T+79T2 1 + 1.72T + 79T^{2}
83 1−15.3T+83T2 1 - 15.3T + 83T^{2}
89 1+7.37T+89T2 1 + 7.37T + 89T^{2}
97 1+7.28T+97T2 1 + 7.28T + 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

−7.47344265592978470499434958217, −6.76160461643818653939943954086, −5.99951104888279126586183068115, −5.41970013391720671249735557688, −4.71732286207248904467726041822, −3.76197200219744052448254976411, −3.06826506186789931603679496117, −2.40748404145750326418098403700, −1.08506239799303951038936861999, 0, 1.08506239799303951038936861999, 2.40748404145750326418098403700, 3.06826506186789931603679496117, 3.76197200219744052448254976411, 4.71732286207248904467726041822, 5.41970013391720671249735557688, 5.99951104888279126586183068115, 6.76160461643818653939943954086, 7.47344265592978470499434958217

Graph of the ZZ-function along the critical line