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Properties

Label 2-4788-1.1-c1-0-41
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  + 0.540·5-s + 7-s + 1.74·11-s − 0.763·13-s + 2.28·17-s − 19-s − 7.40·23-s − 4.70·25-s − 7.94·29-s − 2.76·31-s + 0.540·35-s − 6·37-s + 1.74·41-s − 10.1·43-s − 5.78·47-s + 49-s − 6.19·53-s + 0.944·55-s + 12.6·59-s + 6.94·61-s − 0.412·65-s − 2.47·67-s + 9.69·71-s − 3.52·73-s + 1.74·77-s + 4.94·79-s + 7.27·83-s + ⋯
L(s)  = 1  + 0.241·5-s + 0.377·7-s + 0.527·11-s − 0.211·13-s + 0.554·17-s − 0.229·19-s − 1.54·23-s − 0.941·25-s − 1.47·29-s − 0.496·31-s + 0.0913·35-s − 0.986·37-s + 0.273·41-s − 1.55·43-s − 0.843·47-s + 0.142·49-s − 0.851·53-s + 0.127·55-s + 1.64·59-s + 0.889·61-s − 0.0511·65-s − 0.302·67-s + 1.15·71-s − 0.412·73-s + 0.199·77-s + 0.556·79-s + 0.798·83-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 1−0.540T+5T2 1 - 0.540T + 5T^{2}
11 1−1.74T+11T2 1 - 1.74T + 11T^{2}
13 1+0.763T+13T2 1 + 0.763T + 13T^{2}
17 1−2.28T+17T2 1 - 2.28T + 17T^{2}
23 1+7.40T+23T2 1 + 7.40T + 23T^{2}
29 1+7.94T+29T2 1 + 7.94T + 29T^{2}
31 1+2.76T+31T2 1 + 2.76T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1−1.74T+41T2 1 - 1.74T + 41T^{2}
43 1+10.1T+43T2 1 + 10.1T + 43T^{2}
47 1+5.78T+47T2 1 + 5.78T + 47T^{2}
53 1+6.19T+53T2 1 + 6.19T + 53T^{2}
59 1−12.6T+59T2 1 - 12.6T + 59T^{2}
61 1−6.94T+61T2 1 - 6.94T + 61T^{2}
67 1+2.47T+67T2 1 + 2.47T + 67T^{2}
71 1−9.69T+71T2 1 - 9.69T + 71T^{2}
73 1+3.52T+73T2 1 + 3.52T + 73T^{2}
79 1−4.94T+79T2 1 - 4.94T + 79T^{2}
83 1−7.27T+83T2 1 - 7.27T + 83T^{2}
89 1−1.74T+89T2 1 - 1.74T + 89T^{2}
97 1+17.4T+97T2 1 + 17.4T + 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

−8.017103956451083334039151398849, −7.21762104642214624726940974120, −6.46666093112910879626233499270, −5.67359951482710780346531902426, −5.10838008247957123906965815884, −4.01109552123616620893476015508, −3.52814031667448356876117648730, −2.18296533398540651323550229840, −1.57463874677550758457091818522, 0, 1.57463874677550758457091818522, 2.18296533398540651323550229840, 3.52814031667448356876117648730, 4.01109552123616620893476015508, 5.10838008247957123906965815884, 5.67359951482710780346531902426, 6.46666093112910879626233499270, 7.21762104642214624726940974120, 8.017103956451083334039151398849

Graph of the ZZ-function along the critical line