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Properties

Label 2-8512-1.1-c1-0-103
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  + 2.79·3-s − 3·5-s + 7-s + 4.79·9-s + 3.79·11-s + 13-s − 8.37·15-s + 3.79·17-s − 19-s + 2.79·21-s + 4.58·23-s + 4·25-s + 4.99·27-s − 3.79·29-s + 7.37·31-s + 10.5·33-s − 3·35-s − 5·37-s + 2.79·39-s − 3.79·41-s − 2·43-s − 14.3·45-s + 10.5·47-s + 49-s + 10.5·51-s + 8.37·53-s − 11.3·55-s + ⋯
L(s)  = 1  + 1.61·3-s − 1.34·5-s + 0.377·7-s + 1.59·9-s + 1.14·11-s + 0.277·13-s − 2.16·15-s + 0.919·17-s − 0.229·19-s + 0.609·21-s + 0.955·23-s + 0.800·25-s + 0.962·27-s − 0.704·29-s + 1.32·31-s + 1.84·33-s − 0.507·35-s − 0.821·37-s + 0.446·39-s − 0.592·41-s − 0.304·43-s − 2.14·45-s + 1.54·47-s + 0.142·49-s + 1.48·51-s + 1.15·53-s − 1.53·55-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 3.6940531263.694053126
L(12)L(\frac12) ≈\approx 3.6940531263.694053126
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 1−2.79T+3T2 1 - 2.79T + 3T^{2}
5 1+3T+5T2 1 + 3T + 5T^{2}
11 1−3.79T+11T2 1 - 3.79T + 11T^{2}
13 1−T+13T2 1 - T + 13T^{2}
17 1−3.79T+17T2 1 - 3.79T + 17T^{2}
23 1−4.58T+23T2 1 - 4.58T + 23T^{2}
29 1+3.79T+29T2 1 + 3.79T + 29T^{2}
31 1−7.37T+31T2 1 - 7.37T + 31T^{2}
37 1+5T+37T2 1 + 5T + 37T^{2}
41 1+3.79T+41T2 1 + 3.79T + 41T^{2}
43 1+2T+43T2 1 + 2T + 43T^{2}
47 1−10.5T+47T2 1 - 10.5T + 47T^{2}
53 1−8.37T+53T2 1 - 8.37T + 53T^{2}
59 1+12.1T+59T2 1 + 12.1T + 59T^{2}
61 1−T+61T2 1 - T + 61T^{2}
67 1−9.37T+67T2 1 - 9.37T + 67T^{2}
71 1+12.1T+71T2 1 + 12.1T + 71T^{2}
73 1−16.3T+73T2 1 - 16.3T + 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+14.3T+83T2 1 + 14.3T + 83T^{2}
89 1−7.58T+89T2 1 - 7.58T + 89T^{2}
97 1+7T+97T2 1 + 7T + 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

−7.899059610424442737721966313960, −7.27806321378636324971419931553, −6.82046336853401144973704014953, −5.69062052750856939954129073537, −4.60546983234004483835077327838, −4.03128320830289887892689200710, −3.48273773988637319191263379962, −2.90904162589133298160690701316, −1.81028748186173786243750636227, −0.918274648196623251441518733459, 0.918274648196623251441518733459, 1.81028748186173786243750636227, 2.90904162589133298160690701316, 3.48273773988637319191263379962, 4.03128320830289887892689200710, 4.60546983234004483835077327838, 5.69062052750856939954129073537, 6.82046336853401144973704014953, 7.27806321378636324971419931553, 7.899059610424442737721966313960

Graph of the ZZ-function along the critical line