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Properties

Label 2-4788-1.1-c1-0-39
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  + 0.874·5-s − 7-s + 2.82·11-s + 1.23·13-s − 3.70·17-s + 19-s − 2.82·23-s − 4.23·25-s − 7.19·29-s − 5.70·31-s − 0.874·35-s − 4.47·37-s + 8.48·41-s − 0.763·43-s − 5.45·47-s + 49-s + 4.37·53-s + 2.47·55-s − 14.8·59-s − 12.4·61-s + 1.08·65-s + 11.4·67-s − 3.03·71-s + 6.94·73-s − 2.82·77-s − 1.52·79-s + 4.78·83-s + ⋯
L(s)  = 1  + 0.390·5-s − 0.377·7-s + 0.852·11-s + 0.342·13-s − 0.897·17-s + 0.229·19-s − 0.589·23-s − 0.847·25-s − 1.33·29-s − 1.02·31-s − 0.147·35-s − 0.735·37-s + 1.32·41-s − 0.116·43-s − 0.795·47-s + 0.142·49-s + 0.600·53-s + 0.333·55-s − 1.92·59-s − 1.59·61-s + 0.134·65-s + 1.39·67-s − 0.360·71-s + 0.812·73-s − 0.322·77-s − 0.171·79-s + 0.524·83-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 1−0.874T+5T2 1 - 0.874T + 5T^{2}
11 1−2.82T+11T2 1 - 2.82T + 11T^{2}
13 1−1.23T+13T2 1 - 1.23T + 13T^{2}
17 1+3.70T+17T2 1 + 3.70T + 17T^{2}
23 1+2.82T+23T2 1 + 2.82T + 23T^{2}
29 1+7.19T+29T2 1 + 7.19T + 29T^{2}
31 1+5.70T+31T2 1 + 5.70T + 31T^{2}
37 1+4.47T+37T2 1 + 4.47T + 37T^{2}
41 1−8.48T+41T2 1 - 8.48T + 41T^{2}
43 1+0.763T+43T2 1 + 0.763T + 43T^{2}
47 1+5.45T+47T2 1 + 5.45T + 47T^{2}
53 1−4.37T+53T2 1 - 4.37T + 53T^{2}
59 1+14.8T+59T2 1 + 14.8T + 59T^{2}
61 1+12.4T+61T2 1 + 12.4T + 61T^{2}
67 1−11.4T+67T2 1 - 11.4T + 67T^{2}
71 1+3.03T+71T2 1 + 3.03T + 71T^{2}
73 1−6.94T+73T2 1 - 6.94T + 73T^{2}
79 1+1.52T+79T2 1 + 1.52T + 79T^{2}
83 1−4.78T+83T2 1 - 4.78T + 83T^{2}
89 1−8.48T+89T2 1 - 8.48T + 89T^{2}
97 1+3.52T+97T2 1 + 3.52T + 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

−7.83815003412572286988663583201, −7.20937233504424002828795915379, −6.30924969696008156841077651875, −5.94926427854162062592795687765, −5.00133336429595421938721905192, −4.01259032119415004015264693154, −3.49214986591168343188766404571, −2.27152978849208383081721199037, −1.51033746457637134319415699876, 0, 1.51033746457637134319415699876, 2.27152978849208383081721199037, 3.49214986591168343188766404571, 4.01259032119415004015264693154, 5.00133336429595421938721905192, 5.94926427854162062592795687765, 6.30924969696008156841077651875, 7.20937233504424002828795915379, 7.83815003412572286988663583201

Graph of the ZZ-function along the critical line