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Properties

Label 2-5400-1.1-c1-0-53
Degree 22
Conductor 54005400
Sign −1-1
Analytic cond. 43.119243.1192
Root an. cond. 6.566526.56652
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  − 7-s + 0.449·11-s − 3.89·13-s + 4.89·17-s − 5.89·19-s + 4.44·23-s + 0.449·29-s + 6·31-s − 0.101·37-s − 9.34·41-s + 6·43-s − 4.89·47-s − 6·49-s − 4.44·53-s − 4.89·59-s + 8.79·61-s + 14.7·67-s − 11.5·71-s + 3.89·73-s − 0.449·77-s − 3.89·79-s + 7.55·83-s − 12·89-s + 3.89·91-s − 15.8·97-s − 7.10·101-s − 10.7·103-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.135·11-s − 1.08·13-s + 1.18·17-s − 1.35·19-s + 0.927·23-s + 0.0834·29-s + 1.07·31-s − 0.0166·37-s − 1.45·41-s + 0.914·43-s − 0.714·47-s − 0.857·49-s − 0.611·53-s − 0.637·59-s + 1.12·61-s + 1.80·67-s − 1.37·71-s + 0.456·73-s − 0.0512·77-s − 0.438·79-s + 0.828·83-s − 1.27·89-s + 0.408·91-s − 1.61·97-s − 0.706·101-s − 1.06·103-s + ⋯

Functional equation

Λ(s)=(5400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54005400    =    23⋅33⋅522^{3} \cdot 3^{3} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 43.119243.1192
Root analytic conductor: 6.566526.56652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5400, ( :1/2), −1)(2,\ 5400,\ (\ :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
3 1 1
5 1 1
good7 1+T+7T2 1 + T + 7T^{2}
11 1−0.449T+11T2 1 - 0.449T + 11T^{2}
13 1+3.89T+13T2 1 + 3.89T + 13T^{2}
17 1−4.89T+17T2 1 - 4.89T + 17T^{2}
19 1+5.89T+19T2 1 + 5.89T + 19T^{2}
23 1−4.44T+23T2 1 - 4.44T + 23T^{2}
29 1−0.449T+29T2 1 - 0.449T + 29T^{2}
31 1−6T+31T2 1 - 6T + 31T^{2}
37 1+0.101T+37T2 1 + 0.101T + 37T^{2}
41 1+9.34T+41T2 1 + 9.34T + 41T^{2}
43 1−6T+43T2 1 - 6T + 43T^{2}
47 1+4.89T+47T2 1 + 4.89T + 47T^{2}
53 1+4.44T+53T2 1 + 4.44T + 53T^{2}
59 1+4.89T+59T2 1 + 4.89T + 59T^{2}
61 1−8.79T+61T2 1 - 8.79T + 61T^{2}
67 1−14.7T+67T2 1 - 14.7T + 67T^{2}
71 1+11.5T+71T2 1 + 11.5T + 71T^{2}
73 1−3.89T+73T2 1 - 3.89T + 73T^{2}
79 1+3.89T+79T2 1 + 3.89T + 79T^{2}
83 1−7.55T+83T2 1 - 7.55T + 83T^{2}
89 1+12T+89T2 1 + 12T + 89T^{2}
97 1+15.8T+97T2 1 + 15.8T + 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

−7.963459057594579122403119533656, −6.90653048493958958531445178052, −6.60335726409188197834078815252, −5.58229840793489948713227592113, −4.92810387794930377824471525324, −4.13426275597960821572306435630, −3.18094941228393760994396172663, −2.48212152249939269578907577761, −1.32513231348801711207744572949, 0, 1.32513231348801711207744572949, 2.48212152249939269578907577761, 3.18094941228393760994396172663, 4.13426275597960821572306435630, 4.92810387794930377824471525324, 5.58229840793489948713227592113, 6.60335726409188197834078815252, 6.90653048493958958531445178052, 7.963459057594579122403119533656

Graph of the ZZ-function along the critical line