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Properties

Label 2-8512-1.1-c1-0-12
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  − 2.16·3-s − 4.17·5-s − 7-s + 1.70·9-s + 2.97·11-s − 4.43·13-s + 9.05·15-s + 5.05·17-s − 19-s + 2.16·21-s + 1.98·23-s + 12.4·25-s + 2.80·27-s − 1.00·29-s + 3.05·31-s − 6.44·33-s + 4.17·35-s − 10.6·37-s + 9.61·39-s − 9.85·41-s + 5.68·43-s − 7.12·45-s − 13.1·47-s + 49-s − 10.9·51-s + 10.0·53-s − 12.4·55-s + ⋯
L(s)  = 1  − 1.25·3-s − 1.86·5-s − 0.377·7-s + 0.568·9-s + 0.895·11-s − 1.22·13-s + 2.33·15-s + 1.22·17-s − 0.229·19-s + 0.473·21-s + 0.413·23-s + 2.48·25-s + 0.540·27-s − 0.187·29-s + 0.548·31-s − 1.12·33-s + 0.705·35-s − 1.74·37-s + 1.54·39-s − 1.53·41-s + 0.867·43-s − 1.06·45-s − 1.91·47-s + 0.142·49-s − 1.53·51-s + 1.37·53-s − 1.67·55-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 0.30613362910.3061336291
L(12)L(\frac12) ≈\approx 0.30613362910.3061336291
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+2.16T+3T2 1 + 2.16T + 3T^{2}
5 1+4.17T+5T2 1 + 4.17T + 5T^{2}
11 1−2.97T+11T2 1 - 2.97T + 11T^{2}
13 1+4.43T+13T2 1 + 4.43T + 13T^{2}
17 1−5.05T+17T2 1 - 5.05T + 17T^{2}
23 1−1.98T+23T2 1 - 1.98T + 23T^{2}
29 1+1.00T+29T2 1 + 1.00T + 29T^{2}
31 1−3.05T+31T2 1 - 3.05T + 31T^{2}
37 1+10.6T+37T2 1 + 10.6T + 37T^{2}
41 1+9.85T+41T2 1 + 9.85T + 41T^{2}
43 1−5.68T+43T2 1 - 5.68T + 43T^{2}
47 1+13.1T+47T2 1 + 13.1T + 47T^{2}
53 1−10.0T+53T2 1 - 10.0T + 53T^{2}
59 1+1.75T+59T2 1 + 1.75T + 59T^{2}
61 1−5.57T+61T2 1 - 5.57T + 61T^{2}
67 1+12.8T+67T2 1 + 12.8T + 67T^{2}
71 1+10.4T+71T2 1 + 10.4T + 71T^{2}
73 1+2.97T+73T2 1 + 2.97T + 73T^{2}
79 1+8.52T+79T2 1 + 8.52T + 79T^{2}
83 1+15.8T+83T2 1 + 15.8T + 83T^{2}
89 1+3.74T+89T2 1 + 3.74T + 89T^{2}
97 1−10.3T+97T2 1 - 10.3T + 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

−7.50003713512315809412267016357, −7.11864466793522236685330400282, −6.55855045363588306235594036132, −5.63953768049422939712616523786, −4.93011773784328329825412438058, −4.40113519190492222659876012980, −3.54408350880450614561084097468, −2.97768380500426873244738915300, −1.34561557985656094445373364899, −0.31375759169221242631910114845, 0.31375759169221242631910114845, 1.34561557985656094445373364899, 2.97768380500426873244738915300, 3.54408350880450614561084097468, 4.40113519190492222659876012980, 4.93011773784328329825412438058, 5.63953768049422939712616523786, 6.55855045363588306235594036132, 7.11864466793522236685330400282, 7.50003713512315809412267016357

Graph of the ZZ-function along the critical line