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Properties

Label 2-8512-1.1-c1-0-8
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  + 1.17·3-s − 3.43·5-s − 7-s − 1.61·9-s − 2.61·11-s − 5.43·13-s − 4.04·15-s − 0.611·17-s + 19-s − 1.17·21-s − 1.43·23-s + 6.79·25-s − 5.43·27-s − 1.74·29-s + 0.255·31-s − 3.07·33-s + 3.43·35-s − 5.79·37-s − 6.40·39-s + 8.96·41-s − 7.58·43-s + 5.53·45-s − 10.6·47-s + 49-s − 0.720·51-s − 5.53·53-s + 8.96·55-s + ⋯
L(s)  = 1  + 0.680·3-s − 1.53·5-s − 0.377·7-s − 0.537·9-s − 0.787·11-s − 1.50·13-s − 1.04·15-s − 0.148·17-s + 0.229·19-s − 0.257·21-s − 0.298·23-s + 1.35·25-s − 1.04·27-s − 0.323·29-s + 0.0458·31-s − 0.535·33-s + 0.580·35-s − 0.951·37-s − 1.02·39-s + 1.40·41-s − 1.15·43-s + 0.825·45-s − 1.55·47-s + 0.142·49-s − 0.100·51-s − 0.760·53-s + 1.20·55-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 0.32824281180.3282428118
L(12)L(\frac12) ≈\approx 0.32824281180.3282428118
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 1−1.17T+3T2 1 - 1.17T + 3T^{2}
5 1+3.43T+5T2 1 + 3.43T + 5T^{2}
11 1+2.61T+11T2 1 + 2.61T + 11T^{2}
13 1+5.43T+13T2 1 + 5.43T + 13T^{2}
17 1+0.611T+17T2 1 + 0.611T + 17T^{2}
23 1+1.43T+23T2 1 + 1.43T + 23T^{2}
29 1+1.74T+29T2 1 + 1.74T + 29T^{2}
31 1−0.255T+31T2 1 - 0.255T + 31T^{2}
37 1+5.79T+37T2 1 + 5.79T + 37T^{2}
41 1−8.96T+41T2 1 - 8.96T + 41T^{2}
43 1+7.58T+43T2 1 + 7.58T + 43T^{2}
47 1+10.6T+47T2 1 + 10.6T + 47T^{2}
53 1+5.53T+53T2 1 + 5.53T + 53T^{2}
59 1+4.30T+59T2 1 + 4.30T + 59T^{2}
61 1−1.27T+61T2 1 - 1.27T + 61T^{2}
67 1+0.611T+67T2 1 + 0.611T + 67T^{2}
71 1+1.07T+71T2 1 + 1.07T + 71T^{2}
73 1+11.6T+73T2 1 + 11.6T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1−10.4T+83T2 1 - 10.4T + 83T^{2}
89 1+13.2T+89T2 1 + 13.2T + 89T^{2}
97 1+7.88T+97T2 1 + 7.88T + 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

−7.64395503361347199333710802137, −7.49474124104190865404960196804, −6.61487667750844684262870826751, −5.60194634240205233720156755742, −4.85349783164951712195728069608, −4.20890690505100790948186631897, −3.24900645801641252767418413092, −2.95417738949078819218298199656, −1.97293460850486025101056358784, −0.24985435976138412993749454907, 0.24985435976138412993749454907, 1.97293460850486025101056358784, 2.95417738949078819218298199656, 3.24900645801641252767418413092, 4.20890690505100790948186631897, 4.85349783164951712195728069608, 5.60194634240205233720156755742, 6.61487667750844684262870826751, 7.49474124104190865404960196804, 7.64395503361347199333710802137

Graph of the ZZ-function along the critical line