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Properties

Label 2-4788-1.1-c1-0-1
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  − 3.82·5-s + 7-s − 5.86·11-s − 0.605·13-s + 2.04·17-s + 19-s − 5.86·23-s + 9.60·25-s + 2.04·29-s − 6.60·31-s − 3.82·35-s + 2·37-s − 5.86·41-s − 6.60·43-s − 2.04·47-s + 49-s − 3.82·53-s + 22.4·55-s − 11.7·59-s + 2·61-s + 2.31·65-s − 9.21·67-s + 15.5·71-s + 7.21·73-s − 5.86·77-s − 4·79-s + 3.82·83-s + ⋯
L(s)  = 1  − 1.70·5-s + 0.377·7-s − 1.76·11-s − 0.167·13-s + 0.496·17-s + 0.229·19-s − 1.22·23-s + 1.92·25-s + 0.379·29-s − 1.18·31-s − 0.645·35-s + 0.328·37-s − 0.916·41-s − 1.00·43-s − 0.298·47-s + 0.142·49-s − 0.524·53-s + 3.02·55-s − 1.52·59-s + 0.256·61-s + 0.287·65-s − 1.12·67-s + 1.84·71-s + 0.843·73-s − 0.668·77-s − 0.450·79-s + 0.419·83-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 0.64024868150.6402486815
L(12)L(\frac12) ≈\approx 0.64024868150.6402486815
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+3.82T+5T2 1 + 3.82T + 5T^{2}
11 1+5.86T+11T2 1 + 5.86T + 11T^{2}
13 1+0.605T+13T2 1 + 0.605T + 13T^{2}
17 1−2.04T+17T2 1 - 2.04T + 17T^{2}
23 1+5.86T+23T2 1 + 5.86T + 23T^{2}
29 1−2.04T+29T2 1 - 2.04T + 29T^{2}
31 1+6.60T+31T2 1 + 6.60T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+5.86T+41T2 1 + 5.86T + 41T^{2}
43 1+6.60T+43T2 1 + 6.60T + 43T^{2}
47 1+2.04T+47T2 1 + 2.04T + 47T^{2}
53 1+3.82T+53T2 1 + 3.82T + 53T^{2}
59 1+11.7T+59T2 1 + 11.7T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+9.21T+67T2 1 + 9.21T + 67T^{2}
71 1−15.5T+71T2 1 - 15.5T + 71T^{2}
73 1−7.21T+73T2 1 - 7.21T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−3.82T+83T2 1 - 3.82T + 83T^{2}
89 1−9.41T+89T2 1 - 9.41T + 89T^{2}
97 1+3.21T+97T2 1 + 3.21T + 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

−8.057208645062100914891131107633, −7.72751939041831482213467632678, −7.17461489956016676716570080619, −6.07664026699545060227356977369, −5.09874545954415629192598241149, −4.67995064444232080583014311675, −3.65786179070960783909337212316, −3.09944326336586371046159877865, −1.96667987533243909310714724049, −0.42164205873307662847978819280, 0.42164205873307662847978819280, 1.96667987533243909310714724049, 3.09944326336586371046159877865, 3.65786179070960783909337212316, 4.67995064444232080583014311675, 5.09874545954415629192598241149, 6.07664026699545060227356977369, 7.17461489956016676716570080619, 7.72751939041831482213467632678, 8.057208645062100914891131107633

Graph of the ZZ-function along the critical line