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Properties

Label 2-6e3-1.1-c7-0-11
Degree 22
Conductor 216216
Sign 11
Analytic cond. 67.475167.4751
Root an. cond. 8.214328.21432
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 522.·5-s − 265.·7-s − 4.30e3·11-s + 5.22e3·13-s + 9.35e3·17-s + 5.08e4·19-s − 5.48e4·23-s + 1.94e5·25-s + 1.78e5·29-s − 9.24e4·31-s − 1.38e5·35-s − 4.50e5·37-s + 4.01e5·41-s + 1.87e5·43-s − 3.87e5·47-s − 7.53e5·49-s + 1.05e6·53-s − 2.25e6·55-s − 7.03e5·59-s + 2.53e6·61-s + 2.72e6·65-s + 1.10e6·67-s + 3.82e6·71-s − 1.95e6·73-s + 1.14e6·77-s + 8.46e6·79-s + 1.93e6·83-s + ⋯
L(s)  = 1  + 1.86·5-s − 0.292·7-s − 0.976·11-s + 0.659·13-s + 0.461·17-s + 1.70·19-s − 0.939·23-s + 2.49·25-s + 1.35·29-s − 0.557·31-s − 0.546·35-s − 1.46·37-s + 0.908·41-s + 0.358·43-s − 0.544·47-s − 0.914·49-s + 0.971·53-s − 1.82·55-s − 0.445·59-s + 1.43·61-s + 1.23·65-s + 0.449·67-s + 1.26·71-s − 0.587·73-s + 0.285·77-s + 1.93·79-s + 0.372·83-s + ⋯

Functional equation

Λ(s)=(216s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(216s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 216216    =    23⋅332^{3} \cdot 3^{3}
Sign: 11
Analytic conductor: 67.475167.4751
Root analytic conductor: 8.214328.21432
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 216, ( :7/2), 1)(2,\ 216,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 3.1817676783.181767678
L(12)L(\frac12) ≈\approx 3.1817676783.181767678
L(92)L(\frac{9}{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
good5 1−522.T+7.81e4T2 1 - 522.T + 7.81e4T^{2}
7 1+265.T+8.23e5T2 1 + 265.T + 8.23e5T^{2}
11 1+4.30e3T+1.94e7T2 1 + 4.30e3T + 1.94e7T^{2}
13 1−5.22e3T+6.27e7T2 1 - 5.22e3T + 6.27e7T^{2}
17 1−9.35e3T+4.10e8T2 1 - 9.35e3T + 4.10e8T^{2}
19 1−5.08e4T+8.93e8T2 1 - 5.08e4T + 8.93e8T^{2}
23 1+5.48e4T+3.40e9T2 1 + 5.48e4T + 3.40e9T^{2}
29 1−1.78e5T+1.72e10T2 1 - 1.78e5T + 1.72e10T^{2}
31 1+9.24e4T+2.75e10T2 1 + 9.24e4T + 2.75e10T^{2}
37 1+4.50e5T+9.49e10T2 1 + 4.50e5T + 9.49e10T^{2}
41 1−4.01e5T+1.94e11T2 1 - 4.01e5T + 1.94e11T^{2}
43 1−1.87e5T+2.71e11T2 1 - 1.87e5T + 2.71e11T^{2}
47 1+3.87e5T+5.06e11T2 1 + 3.87e5T + 5.06e11T^{2}
53 1−1.05e6T+1.17e12T2 1 - 1.05e6T + 1.17e12T^{2}
59 1+7.03e5T+2.48e12T2 1 + 7.03e5T + 2.48e12T^{2}
61 1−2.53e6T+3.14e12T2 1 - 2.53e6T + 3.14e12T^{2}
67 1−1.10e6T+6.06e12T2 1 - 1.10e6T + 6.06e12T^{2}
71 1−3.82e6T+9.09e12T2 1 - 3.82e6T + 9.09e12T^{2}
73 1+1.95e6T+1.10e13T2 1 + 1.95e6T + 1.10e13T^{2}
79 1−8.46e6T+1.92e13T2 1 - 8.46e6T + 1.92e13T^{2}
83 1−1.93e6T+2.71e13T2 1 - 1.93e6T + 2.71e13T^{2}
89 1+2.68e6T+4.42e13T2 1 + 2.68e6T + 4.42e13T^{2}
97 1−4.35e6T+8.07e13T2 1 - 4.35e6T + 8.07e13T^{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

−10.74171049194771486507493653240, −10.00627244303562048186991808883, −9.355145047736504987633839846075, −8.147405732479905932969936499301, −6.80151308716602436935356049926, −5.77104052499823986921611492930, −5.15297758630648953135467588763, −3.24854316226546024917919458372, −2.15479516993937576637358886634, −0.980957991107469459765747449631, 0.980957991107469459765747449631, 2.15479516993937576637358886634, 3.24854316226546024917919458372, 5.15297758630648953135467588763, 5.77104052499823986921611492930, 6.80151308716602436935356049926, 8.147405732479905932969936499301, 9.355145047736504987633839846075, 10.00627244303562048186991808883, 10.74171049194771486507493653240

Graph of the ZZ-function along the critical line