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Properties

Label 2-4788-1.1-c1-0-20
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  + 5-s + 7-s + 5.09·11-s + 2.23·13-s + 5.38·17-s − 19-s − 4.70·23-s − 4·25-s + 5.85·29-s − 4.61·31-s + 35-s + 6.70·37-s + 11.0·41-s − 0.472·43-s + 8.70·47-s + 49-s − 1.32·53-s + 5.09·55-s − 2.70·59-s − 3.47·61-s + 2.23·65-s + 9.09·67-s − 11.1·71-s − 12.0·73-s + 5.09·77-s + 10.9·79-s − 12.3·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.53·11-s + 0.620·13-s + 1.30·17-s − 0.229·19-s − 0.981·23-s − 0.800·25-s + 1.08·29-s − 0.829·31-s + 0.169·35-s + 1.10·37-s + 1.73·41-s − 0.0720·43-s + 1.27·47-s + 0.142·49-s − 0.182·53-s + 0.686·55-s − 0.352·59-s − 0.444·61-s + 0.277·65-s + 1.11·67-s − 1.32·71-s − 1.41·73-s + 0.580·77-s + 1.23·79-s − 1.35·83-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 2.7553822432.755382243
L(12)L(\frac12) ≈\approx 2.7553822432.755382243
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−T+5T2 1 - T + 5T^{2}
11 1−5.09T+11T2 1 - 5.09T + 11T^{2}
13 1−2.23T+13T2 1 - 2.23T + 13T^{2}
17 1−5.38T+17T2 1 - 5.38T + 17T^{2}
23 1+4.70T+23T2 1 + 4.70T + 23T^{2}
29 1−5.85T+29T2 1 - 5.85T + 29T^{2}
31 1+4.61T+31T2 1 + 4.61T + 31T^{2}
37 1−6.70T+37T2 1 - 6.70T + 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1+0.472T+43T2 1 + 0.472T + 43T^{2}
47 1−8.70T+47T2 1 - 8.70T + 47T^{2}
53 1+1.32T+53T2 1 + 1.32T + 53T^{2}
59 1+2.70T+59T2 1 + 2.70T + 59T^{2}
61 1+3.47T+61T2 1 + 3.47T + 61T^{2}
67 1−9.09T+67T2 1 - 9.09T + 67T^{2}
71 1+11.1T+71T2 1 + 11.1T + 71T^{2}
73 1+12.0T+73T2 1 + 12.0T + 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+12.3T+83T2 1 + 12.3T + 83T^{2}
89 1+10.1T+89T2 1 + 10.1T + 89T^{2}
97 1+4.70T+97T2 1 + 4.70T + 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

−8.236908792790018649354967695043, −7.64857668808539261312923307109, −6.77011546144461102915573521675, −5.92320012637521954776406096801, −5.68974078030059396919080935511, −4.32098599213049420890103892891, −3.95472121609748790896954782595, −2.86869788755854741994298255701, −1.75137314821791586861668175608, −1.00784907577002794754355377683, 1.00784907577002794754355377683, 1.75137314821791586861668175608, 2.86869788755854741994298255701, 3.95472121609748790896954782595, 4.32098599213049420890103892891, 5.68974078030059396919080935511, 5.92320012637521954776406096801, 6.77011546144461102915573521675, 7.64857668808539261312923307109, 8.236908792790018649354967695043

Graph of the ZZ-function along the critical line