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Properties

Label 2-4788-1.1-c1-0-32
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  − 1.42·5-s − 7-s + 3.27·11-s + 3.42·13-s − 5.27·17-s − 19-s − 0.574·23-s − 2.96·25-s + 0.122·29-s + 2.12·31-s + 1.42·35-s − 3.96·37-s + 4.66·41-s − 11.9·43-s − 3.11·47-s + 49-s + 6.08·53-s − 4.66·55-s + 1.72·59-s + 12.2·61-s − 4.88·65-s − 5.27·67-s + 6.81·71-s − 11.5·73-s − 3.27·77-s + 11.0·79-s − 5.23·83-s + ⋯
L(s)  = 1  − 0.637·5-s − 0.377·7-s + 0.986·11-s + 0.950·13-s − 1.27·17-s − 0.229·19-s − 0.119·23-s − 0.593·25-s + 0.0227·29-s + 0.381·31-s + 0.241·35-s − 0.652·37-s + 0.728·41-s − 1.81·43-s − 0.454·47-s + 0.142·49-s + 0.836·53-s − 0.628·55-s + 0.224·59-s + 1.56·61-s − 0.605·65-s − 0.643·67-s + 0.809·71-s − 1.35·73-s − 0.372·77-s + 1.24·79-s − 0.574·83-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+1.42T+5T2 1 + 1.42T + 5T^{2}
11 1−3.27T+11T2 1 - 3.27T + 11T^{2}
13 1−3.42T+13T2 1 - 3.42T + 13T^{2}
17 1+5.27T+17T2 1 + 5.27T + 17T^{2}
23 1+0.574T+23T2 1 + 0.574T + 23T^{2}
29 1−0.122T+29T2 1 - 0.122T + 29T^{2}
31 1−2.12T+31T2 1 - 2.12T + 31T^{2}
37 1+3.96T+37T2 1 + 3.96T + 37T^{2}
41 1−4.66T+41T2 1 - 4.66T + 41T^{2}
43 1+11.9T+43T2 1 + 11.9T + 43T^{2}
47 1+3.11T+47T2 1 + 3.11T + 47T^{2}
53 1−6.08T+53T2 1 - 6.08T + 53T^{2}
59 1−1.72T+59T2 1 - 1.72T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1+5.27T+67T2 1 + 5.27T + 67T^{2}
71 1−6.81T+71T2 1 - 6.81T + 71T^{2}
73 1+11.5T+73T2 1 + 11.5T + 73T^{2}
79 1−11.0T+79T2 1 - 11.0T + 79T^{2}
83 1+5.23T+83T2 1 + 5.23T + 83T^{2}
89 1−1.45T+89T2 1 - 1.45T + 89T^{2}
97 1−17.6T+97T2 1 - 17.6T + 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

−8.082853725293927203213709871720, −7.01460571540713490223087234466, −6.58981154981396501174413316534, −5.89121501983364901012169717262, −4.84329458615620407145141925152, −3.95715600855877940973954997520, −3.61417168403294593911068923119, −2.40920811542236548713162993931, −1.32804388299010437186613276852, 0, 1.32804388299010437186613276852, 2.40920811542236548713162993931, 3.61417168403294593911068923119, 3.95715600855877940973954997520, 4.84329458615620407145141925152, 5.89121501983364901012169717262, 6.58981154981396501174413316534, 7.01460571540713490223087234466, 8.082853725293927203213709871720

Graph of the ZZ-function along the critical line