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Properties

Label 2-4788-1.1-c1-0-26
Degree 22
Conductor 47884788
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.23·5-s − 7-s + 2.61·11-s + 3.47·13-s + 5.85·17-s + 19-s + 8.23·23-s + 12.9·25-s + 4.61·29-s − 3.85·31-s − 4.23·35-s − 8.23·37-s − 11.5·41-s + 4.47·43-s − 7.47·47-s + 49-s − 6.09·53-s + 11.0·55-s − 11·59-s − 4.23·61-s + 14.7·65-s − 7.56·67-s + 3.76·71-s + 3.61·73-s − 2.61·77-s − 4.47·79-s − 10.5·83-s + ⋯
L(s)  = 1  + 1.89·5-s − 0.377·7-s + 0.789·11-s + 0.962·13-s + 1.41·17-s + 0.229·19-s + 1.71·23-s + 2.58·25-s + 0.857·29-s − 0.692·31-s − 0.716·35-s − 1.35·37-s − 1.80·41-s + 0.681·43-s − 1.08·47-s + 0.142·49-s − 0.836·53-s + 1.49·55-s − 1.43·59-s − 0.542·61-s + 1.82·65-s − 0.923·67-s + 0.446·71-s + 0.423·73-s − 0.298·77-s − 0.503·79-s − 1.15·83-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: 11
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: 00
Selberg data: (2, 4788, ( :1/2), 1)(2,\ 4788,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4208147163.420814716
L(12)L(\frac12) ≈\approx 3.4208147163.420814716
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 1−4.23T+5T2 1 - 4.23T + 5T^{2}
11 1−2.61T+11T2 1 - 2.61T + 11T^{2}
13 1−3.47T+13T2 1 - 3.47T + 13T^{2}
17 1−5.85T+17T2 1 - 5.85T + 17T^{2}
23 1−8.23T+23T2 1 - 8.23T + 23T^{2}
29 1−4.61T+29T2 1 - 4.61T + 29T^{2}
31 1+3.85T+31T2 1 + 3.85T + 31T^{2}
37 1+8.23T+37T2 1 + 8.23T + 37T^{2}
41 1+11.5T+41T2 1 + 11.5T + 41T^{2}
43 1−4.47T+43T2 1 - 4.47T + 43T^{2}
47 1+7.47T+47T2 1 + 7.47T + 47T^{2}
53 1+6.09T+53T2 1 + 6.09T + 53T^{2}
59 1+11T+59T2 1 + 11T + 59T^{2}
61 1+4.23T+61T2 1 + 4.23T + 61T^{2}
67 1+7.56T+67T2 1 + 7.56T + 67T^{2}
71 1−3.76T+71T2 1 - 3.76T + 71T^{2}
73 1−3.61T+73T2 1 - 3.61T + 73T^{2}
79 1+4.47T+79T2 1 + 4.47T + 79T^{2}
83 1+10.5T+83T2 1 + 10.5T + 83T^{2}
89 1−15.7T+89T2 1 - 15.7T + 89T^{2}
97 1+3.47T+97T2 1 + 3.47T + 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

−8.587287418001297337183531412402, −7.40130183663487694173232031749, −6.56471656679738190925103441178, −6.22370627862026535303519004543, −5.39045758100929636203926446832, −4.88233114689324812183368757222, −3.44019574146790754690723858989, −2.99936024936045281814387860215, −1.66296625781287452626093035839, −1.19450046447890827961442499760, 1.19450046447890827961442499760, 1.66296625781287452626093035839, 2.99936024936045281814387860215, 3.44019574146790754690723858989, 4.88233114689324812183368757222, 5.39045758100929636203926446832, 6.22370627862026535303519004543, 6.56471656679738190925103441178, 7.40130183663487694173232031749, 8.587287418001297337183531412402

Graph of the ZZ-function along the critical line