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Properties

Label 2-4788-1.1-c1-0-37
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  + 2.73·5-s − 7-s − 3.46·11-s − 4·13-s − 4.19·17-s − 19-s + 7.46·23-s + 2.46·25-s + 6.19·29-s + 4.92·31-s − 2.73·35-s − 7.46·37-s + 9.46·41-s − 3.46·43-s − 2.73·47-s + 49-s − 8.73·53-s − 9.46·55-s + 10.9·59-s − 14.3·61-s − 10.9·65-s + 1.46·67-s − 10.1·71-s − 15.8·73-s + 3.46·77-s − 13.4·79-s + 1.26·83-s + ⋯
L(s)  = 1  + 1.22·5-s − 0.377·7-s − 1.04·11-s − 1.10·13-s − 1.01·17-s − 0.229·19-s + 1.55·23-s + 0.492·25-s + 1.15·29-s + 0.885·31-s − 0.461·35-s − 1.22·37-s + 1.47·41-s − 0.528·43-s − 0.398·47-s + 0.142·49-s − 1.19·53-s − 1.27·55-s + 1.42·59-s − 1.84·61-s − 1.35·65-s + 0.178·67-s − 1.21·71-s − 1.85·73-s + 0.394·77-s − 1.51·79-s + 0.139·83-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 1−2.73T+5T2 1 - 2.73T + 5T^{2}
11 1+3.46T+11T2 1 + 3.46T + 11T^{2}
13 1+4T+13T2 1 + 4T + 13T^{2}
17 1+4.19T+17T2 1 + 4.19T + 17T^{2}
23 1−7.46T+23T2 1 - 7.46T + 23T^{2}
29 1−6.19T+29T2 1 - 6.19T + 29T^{2}
31 1−4.92T+31T2 1 - 4.92T + 31T^{2}
37 1+7.46T+37T2 1 + 7.46T + 37T^{2}
41 1−9.46T+41T2 1 - 9.46T + 41T^{2}
43 1+3.46T+43T2 1 + 3.46T + 43T^{2}
47 1+2.73T+47T2 1 + 2.73T + 47T^{2}
53 1+8.73T+53T2 1 + 8.73T + 53T^{2}
59 1−10.9T+59T2 1 - 10.9T + 59T^{2}
61 1+14.3T+61T2 1 + 14.3T + 61T^{2}
67 1−1.46T+67T2 1 - 1.46T + 67T^{2}
71 1+10.1T+71T2 1 + 10.1T + 71T^{2}
73 1+15.8T+73T2 1 + 15.8T + 73T^{2}
79 1+13.4T+79T2 1 + 13.4T + 79T^{2}
83 1−1.26T+83T2 1 - 1.26T + 83T^{2}
89 1+9.46T+89T2 1 + 9.46T + 89T^{2}
97 1+6T+97T2 1 + 6T + 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

−7.905762761133778948798963994776, −7.04712134720812364061126020026, −6.52335188093155764935448228034, −5.70424029392471525402457468158, −5.00147348262843879862010636695, −4.43207112370119232712953663037, −2.84220825316301115835751353962, −2.64715394159476842232378922552, −1.49900195174847536244542047421, 0, 1.49900195174847536244542047421, 2.64715394159476842232378922552, 2.84220825316301115835751353962, 4.43207112370119232712953663037, 5.00147348262843879862010636695, 5.70424029392471525402457468158, 6.52335188093155764935448228034, 7.04712134720812364061126020026, 7.905762761133778948798963994776

Graph of the ZZ-function along the critical line