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Properties

Label 2-4788-1.1-c1-0-6
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  − 0.236·5-s − 7-s + 0.381·11-s − 5.47·13-s − 0.854·17-s + 19-s + 3.76·23-s − 4.94·25-s + 2.38·29-s + 2.85·31-s + 0.236·35-s − 3.76·37-s + 8.56·41-s − 4.47·43-s + 1.47·47-s + 49-s + 5.09·53-s − 0.0901·55-s − 11·59-s + 0.236·61-s + 1.29·65-s + 12.5·67-s + 8.23·71-s + 1.38·73-s − 0.381·77-s + 4.47·79-s + 9.56·83-s + ⋯
L(s)  = 1  − 0.105·5-s − 0.377·7-s + 0.115·11-s − 1.51·13-s − 0.207·17-s + 0.229·19-s + 0.784·23-s − 0.988·25-s + 0.442·29-s + 0.512·31-s + 0.0399·35-s − 0.618·37-s + 1.33·41-s − 0.681·43-s + 0.214·47-s + 0.142·49-s + 0.699·53-s − 0.0121·55-s − 1.43·59-s + 0.0302·61-s + 0.160·65-s + 1.53·67-s + 0.977·71-s + 0.161·73-s − 0.0435·77-s + 0.503·79-s + 1.04·83-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 1.4041395471.404139547
L(12)L(\frac12) ≈\approx 1.4041395471.404139547
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+0.236T+5T2 1 + 0.236T + 5T^{2}
11 1−0.381T+11T2 1 - 0.381T + 11T^{2}
13 1+5.47T+13T2 1 + 5.47T + 13T^{2}
17 1+0.854T+17T2 1 + 0.854T + 17T^{2}
23 1−3.76T+23T2 1 - 3.76T + 23T^{2}
29 1−2.38T+29T2 1 - 2.38T + 29T^{2}
31 1−2.85T+31T2 1 - 2.85T + 31T^{2}
37 1+3.76T+37T2 1 + 3.76T + 37T^{2}
41 1−8.56T+41T2 1 - 8.56T + 41T^{2}
43 1+4.47T+43T2 1 + 4.47T + 43T^{2}
47 1−1.47T+47T2 1 - 1.47T + 47T^{2}
53 1−5.09T+53T2 1 - 5.09T + 53T^{2}
59 1+11T+59T2 1 + 11T + 59T^{2}
61 1−0.236T+61T2 1 - 0.236T + 61T^{2}
67 1−12.5T+67T2 1 - 12.5T + 67T^{2}
71 1−8.23T+71T2 1 - 8.23T + 71T^{2}
73 1−1.38T+73T2 1 - 1.38T + 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1−9.56T+83T2 1 - 9.56T + 83T^{2}
89 1−2.29T+89T2 1 - 2.29T + 89T^{2}
97 1−5.47T+97T2 1 - 5.47T + 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

−8.187609483925142055726272939078, −7.52566956100109919850461449398, −6.90336896389729062104301312405, −6.17973370421833969913337415050, −5.25393581638305661136394867434, −4.65537891262678610530872460445, −3.73559602616866165646140914308, −2.83456804697838287827206118048, −2.04883608797897585665717203348, −0.63304334015876680491873545576, 0.63304334015876680491873545576, 2.04883608797897585665717203348, 2.83456804697838287827206118048, 3.73559602616866165646140914308, 4.65537891262678610530872460445, 5.25393581638305661136394867434, 6.17973370421833969913337415050, 6.90336896389729062104301312405, 7.52566956100109919850461449398, 8.187609483925142055726272939078

Graph of the ZZ-function along the critical line