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Properties

Label 2-4788-1.1-c1-0-8
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  − 1.23·5-s + 7-s + 2·11-s − 4.47·13-s + 1.23·17-s − 19-s + 2·23-s − 3.47·25-s + 7.23·29-s + 4·31-s − 1.23·35-s + 4.47·37-s − 6.47·41-s − 10.4·43-s + 9.23·47-s + 49-s − 12.1·53-s − 2.47·55-s + 12.9·59-s + 0.472·61-s + 5.52·65-s + 4.94·67-s + 7.23·71-s + 6·73-s + 2·77-s − 16.9·79-s + 6.76·83-s + ⋯
L(s)  = 1  − 0.552·5-s + 0.377·7-s + 0.603·11-s − 1.24·13-s + 0.299·17-s − 0.229·19-s + 0.417·23-s − 0.694·25-s + 1.34·29-s + 0.718·31-s − 0.208·35-s + 0.735·37-s − 1.01·41-s − 1.59·43-s + 1.34·47-s + 0.142·49-s − 1.67·53-s − 0.333·55-s + 1.68·59-s + 0.0604·61-s + 0.685·65-s + 0.604·67-s + 0.858·71-s + 0.702·73-s + 0.227·77-s − 1.90·79-s + 0.742·83-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 1.6357923681.635792368
L(12)L(\frac12) ≈\approx 1.6357923681.635792368
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+1.23T+5T2 1 + 1.23T + 5T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+4.47T+13T2 1 + 4.47T + 13T^{2}
17 1−1.23T+17T2 1 - 1.23T + 17T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1−7.23T+29T2 1 - 7.23T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−4.47T+37T2 1 - 4.47T + 37T^{2}
41 1+6.47T+41T2 1 + 6.47T + 41T^{2}
43 1+10.4T+43T2 1 + 10.4T + 43T^{2}
47 1−9.23T+47T2 1 - 9.23T + 47T^{2}
53 1+12.1T+53T2 1 + 12.1T + 53T^{2}
59 1−12.9T+59T2 1 - 12.9T + 59T^{2}
61 1−0.472T+61T2 1 - 0.472T + 61T^{2}
67 1−4.94T+67T2 1 - 4.94T + 67T^{2}
71 1−7.23T+71T2 1 - 7.23T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+16.9T+79T2 1 + 16.9T + 79T^{2}
83 1−6.76T+83T2 1 - 6.76T + 83T^{2}
89 1−1.52T+89T2 1 - 1.52T + 89T^{2}
97 1−16.4T+97T2 1 - 16.4T + 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

−8.230543230308003743596717645668, −7.61685736823612803634141047484, −6.88209087899201275699280465783, −6.23821385509782428254031713322, −5.14304323514114182231205580808, −4.64246187342294396653448507157, −3.80244999295771040082931127484, −2.89864604317108997233748518703, −1.93521439172926010742921922986, −0.70442958664710639496682572003, 0.70442958664710639496682572003, 1.93521439172926010742921922986, 2.89864604317108997233748518703, 3.80244999295771040082931127484, 4.64246187342294396653448507157, 5.14304323514114182231205580808, 6.23821385509782428254031713322, 6.88209087899201275699280465783, 7.61685736823612803634141047484, 8.230543230308003743596717645668

Graph of the ZZ-function along the critical line