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Properties

Label 2-4732-1.1-c1-0-15
Degree 22
Conductor 47324732
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.25·3-s − 4.27·5-s − 7-s + 2.08·9-s − 5.44·11-s + 9.64·15-s − 6.61·17-s + 2.49·19-s + 2.25·21-s + 5.96·23-s + 13.2·25-s + 2.05·27-s − 0.282·29-s + 1.74·31-s + 12.2·33-s + 4.27·35-s + 3.60·37-s − 3.74·41-s + 3.43·43-s − 8.93·45-s − 1.20·47-s + 49-s + 14.9·51-s + 8.16·53-s + 23.2·55-s − 5.63·57-s − 3.16·59-s + ⋯
L(s)  = 1  − 1.30·3-s − 1.91·5-s − 0.377·7-s + 0.696·9-s − 1.64·11-s + 2.49·15-s − 1.60·17-s + 0.573·19-s + 0.492·21-s + 1.24·23-s + 2.65·25-s + 0.395·27-s − 0.0524·29-s + 0.313·31-s + 2.13·33-s + 0.722·35-s + 0.592·37-s − 0.585·41-s + 0.523·43-s − 1.33·45-s − 0.175·47-s + 0.142·49-s + 2.09·51-s + 1.12·53-s + 3.13·55-s − 0.746·57-s − 0.411·59-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: −1-1
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: 11
Selberg data: (2, 4732, ( :1/2), −1)(2,\ 4732,\ (\ :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
13 1 1
good3 1+2.25T+3T2 1 + 2.25T + 3T^{2}
5 1+4.27T+5T2 1 + 4.27T + 5T^{2}
11 1+5.44T+11T2 1 + 5.44T + 11T^{2}
17 1+6.61T+17T2 1 + 6.61T + 17T^{2}
19 1−2.49T+19T2 1 - 2.49T + 19T^{2}
23 1−5.96T+23T2 1 - 5.96T + 23T^{2}
29 1+0.282T+29T2 1 + 0.282T + 29T^{2}
31 1−1.74T+31T2 1 - 1.74T + 31T^{2}
37 1−3.60T+37T2 1 - 3.60T + 37T^{2}
41 1+3.74T+41T2 1 + 3.74T + 41T^{2}
43 1−3.43T+43T2 1 - 3.43T + 43T^{2}
47 1+1.20T+47T2 1 + 1.20T + 47T^{2}
53 1−8.16T+53T2 1 - 8.16T + 53T^{2}
59 1+3.16T+59T2 1 + 3.16T + 59T^{2}
61 1+4.96T+61T2 1 + 4.96T + 61T^{2}
67 1+10.9T+67T2 1 + 10.9T + 67T^{2}
71 1+5.57T+71T2 1 + 5.57T + 71T^{2}
73 1−4.66T+73T2 1 - 4.66T + 73T^{2}
79 1−9.40T+79T2 1 - 9.40T + 79T^{2}
83 1−0.00717T+83T2 1 - 0.00717T + 83T^{2}
89 1−17.1T+89T2 1 - 17.1T + 89T^{2}
97 1+3.20T+97T2 1 + 3.20T + 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

−7.77993467617956224259490231513, −7.17881114518498019854424860058, −6.61118831043964808513638033182, −5.65273112527185756122398564369, −4.80444443850808483107478611469, −4.51737423823296718291984516352, −3.37818971916665580215450447009, −2.63162730664729171674134737660, −0.73414796206686743725963938960, 0, 0.73414796206686743725963938960, 2.63162730664729171674134737660, 3.37818971916665580215450447009, 4.51737423823296718291984516352, 4.80444443850808483107478611469, 5.65273112527185756122398564369, 6.61118831043964808513638033182, 7.17881114518498019854424860058, 7.77993467617956224259490231513

Graph of the ZZ-function along the critical line