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Properties

Label 2-4732-1.1-c1-0-42
Degree 22
Conductor 47324732
Sign 11
Analytic cond. 37.785237.7852
Root an. cond. 6.146966.14696
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  + 1.79·3-s + 3.79·5-s − 7-s + 0.208·9-s + 3.79·11-s + 6.79·15-s + 3·17-s − 7.37·19-s − 1.79·21-s + 6·23-s + 9.37·25-s − 5.00·27-s − 2.20·29-s + 31-s + 6.79·33-s − 3.79·35-s + 4·37-s + 1.58·41-s + 4.37·43-s + 0.791·45-s + 12.1·47-s + 49-s + 5.37·51-s − 10.5·53-s + 14.3·55-s − 13.2·57-s + 9·59-s + ⋯
L(s)  = 1  + 1.03·3-s + 1.69·5-s − 0.377·7-s + 0.0695·9-s + 1.14·11-s + 1.75·15-s + 0.727·17-s − 1.69·19-s − 0.390·21-s + 1.25·23-s + 1.87·25-s − 0.962·27-s − 0.410·29-s + 0.179·31-s + 1.18·33-s − 0.640·35-s + 0.657·37-s + 0.247·41-s + 0.667·43-s + 0.117·45-s + 1.77·47-s + 0.142·49-s + 0.752·51-s − 1.45·53-s + 1.93·55-s − 1.74·57-s + 1.17·59-s + ⋯

Functional equation

Λ(s)=(4732s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4732s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47324732    =    22⋅7⋅1322^{2} \cdot 7 \cdot 13^{2}
Sign: 11
Analytic conductor: 37.785237.7852
Root analytic conductor: 6.146966.14696
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4732, ( :1/2), 1)(2,\ 4732,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0952303484.095230348
L(12)L(\frac12) ≈\approx 4.0952303484.095230348
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
13 1 1
good3 1−1.79T+3T2 1 - 1.79T + 3T^{2}
5 1−3.79T+5T2 1 - 3.79T + 5T^{2}
11 1−3.79T+11T2 1 - 3.79T + 11T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+7.37T+19T2 1 + 7.37T + 19T^{2}
23 1−6T+23T2 1 - 6T + 23T^{2}
29 1+2.20T+29T2 1 + 2.20T + 29T^{2}
31 1−T+31T2 1 - T + 31T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1−1.58T+41T2 1 - 1.58T + 41T^{2}
43 1−4.37T+43T2 1 - 4.37T + 43T^{2}
47 1−12.1T+47T2 1 - 12.1T + 47T^{2}
53 1+10.5T+53T2 1 + 10.5T + 53T^{2}
59 1−9T+59T2 1 - 9T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−7T+67T2 1 - 7T + 67T^{2}
71 1+7.58T+71T2 1 + 7.58T + 71T^{2}
73 1+14T+73T2 1 + 14T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−9T+83T2 1 - 9T + 83T^{2}
89 1−5.37T+89T2 1 - 5.37T + 89T^{2}
97 1−18.3T+97T2 1 - 18.3T + 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

−8.632164310327843624409446583232, −7.59341398981278110193384535627, −6.74100028685390125904658899133, −6.12551863375761642811233815104, −5.58224591100784967254643811304, −4.48098865359157370573716348482, −3.59305087609701513458108466387, −2.69726886461902622718065349987, −2.11474084394772602895127858407, −1.14728394116008939265892258862, 1.14728394116008939265892258862, 2.11474084394772602895127858407, 2.69726886461902622718065349987, 3.59305087609701513458108466387, 4.48098865359157370573716348482, 5.58224591100784967254643811304, 6.12551863375761642811233815104, 6.74100028685390125904658899133, 7.59341398981278110193384535627, 8.632164310327843624409446583232

Graph of the ZZ-function along the critical line