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Properties

Label 2-4788-1.1-c1-0-30
Degree 22
Conductor 47884788
Sign −1-1
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  − 3·5-s + 7-s + 3.79·11-s − 13-s − 3.79·17-s + 19-s − 4.58·23-s + 4·25-s − 3.79·29-s + 7.37·31-s − 3·35-s + 5·37-s + 3.79·41-s + 2·43-s − 10.5·47-s + 49-s + 8.37·53-s − 11.3·55-s − 12.1·59-s − 61-s + 3·65-s − 9.37·67-s + 12.1·71-s + 16.3·73-s + 3.79·77-s − 10·79-s − 14.3·83-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 1.14·11-s − 0.277·13-s − 0.919·17-s + 0.229·19-s − 0.955·23-s + 0.800·25-s − 0.704·29-s + 1.32·31-s − 0.507·35-s + 0.821·37-s + 0.592·41-s + 0.304·43-s − 1.54·47-s + 0.142·49-s + 1.15·53-s − 1.53·55-s − 1.58·59-s − 0.128·61-s + 0.372·65-s − 1.14·67-s + 1.44·71-s + 1.91·73-s + 0.432·77-s − 1.12·79-s − 1.57·83-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4788, ( :1/2), −1)(2,\ 4788,\ (\ :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
7 1−T 1 - T
19 1−T 1 - T
good5 1+3T+5T2 1 + 3T + 5T^{2}
11 1−3.79T+11T2 1 - 3.79T + 11T^{2}
13 1+T+13T2 1 + T + 13T^{2}
17 1+3.79T+17T2 1 + 3.79T + 17T^{2}
23 1+4.58T+23T2 1 + 4.58T + 23T^{2}
29 1+3.79T+29T2 1 + 3.79T + 29T^{2}
31 1−7.37T+31T2 1 - 7.37T + 31T^{2}
37 1−5T+37T2 1 - 5T + 37T^{2}
41 1−3.79T+41T2 1 - 3.79T + 41T^{2}
43 1−2T+43T2 1 - 2T + 43T^{2}
47 1+10.5T+47T2 1 + 10.5T + 47T^{2}
53 1−8.37T+53T2 1 - 8.37T + 53T^{2}
59 1+12.1T+59T2 1 + 12.1T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1+9.37T+67T2 1 + 9.37T + 67T^{2}
71 1−12.1T+71T2 1 - 12.1T + 71T^{2}
73 1−16.3T+73T2 1 - 16.3T + 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+14.3T+83T2 1 + 14.3T + 83T^{2}
89 1+7.58T+89T2 1 + 7.58T + 89T^{2}
97 1+7T+97T2 1 + 7T + 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

−8.019719056045307954021325888623, −7.25314307276747735944767511874, −6.59998835000921810157139539036, −5.81301493208881776699562282671, −4.62729378536117130988037843585, −4.23876214820757577371486467760, −3.52317966597678152123509557470, −2.44794237707911530887908187109, −1.26796192541073129789232848734, 0, 1.26796192541073129789232848734, 2.44794237707911530887908187109, 3.52317966597678152123509557470, 4.23876214820757577371486467760, 4.62729378536117130988037843585, 5.81301493208881776699562282671, 6.59998835000921810157139539036, 7.25314307276747735944767511874, 8.019719056045307954021325888623

Graph of the ZZ-function along the critical line