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Properties

Label 2-3e3-3.2-c6-0-3
Degree 22
Conductor 2727
Sign 11
Analytic cond. 6.211466.21146
Root an. cond. 2.492282.49228
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6i·2-s + 28·4-s + 240i·5-s + 299·7-s − 552i·8-s + 1.44e3·10-s + 624i·11-s + 2.49e3·13-s − 1.79e3i·14-s − 1.52e3·16-s − 1.87e3i·17-s − 2.50e3·19-s + 6.72e3i·20-s + 3.74e3·22-s + 1.43e4i·23-s + ⋯
L(s)  = 1  − 0.750i·2-s + 0.437·4-s + 1.91i·5-s + 0.871·7-s − 1.07i·8-s + 1.43·10-s + 0.468i·11-s + 1.13·13-s − 0.653i·14-s − 0.371·16-s − 0.381i·17-s − 0.365·19-s + 0.839i·20-s + 0.351·22-s + 1.17i·23-s + ⋯

Functional equation

Λ(s)=(27s/2 ΓC(s) L(s)=( Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(27s/2 ΓC(s+3) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2727    =    333^{3}
Sign: 11
Analytic conductor: 6.211466.21146
Root analytic conductor: 2.492282.49228
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ27(26,⋅)\chi_{27} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 27, ( :3), 1)(2,\ 27,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.969921.96992
L(12)L(\frac12) ≈\approx 1.969921.96992
L(4)L(4) 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)
bad3 1 1
good2 1+6iT−64T2 1 + 6iT - 64T^{2}
5 1−240iT−1.56e4T2 1 - 240iT - 1.56e4T^{2}
7 1−299T+1.17e5T2 1 - 299T + 1.17e5T^{2}
11 1−624iT−1.77e6T2 1 - 624iT - 1.77e6T^{2}
13 1−2.49e3T+4.82e6T2 1 - 2.49e3T + 4.82e6T^{2}
17 1+1.87e3iT−2.41e7T2 1 + 1.87e3iT - 2.41e7T^{2}
19 1+2.50e3T+4.70e7T2 1 + 2.50e3T + 4.70e7T^{2}
23 1−1.43e4iT−1.48e8T2 1 - 1.43e4iT - 1.48e8T^{2}
29 1+2.37e4iT−5.94e8T2 1 + 2.37e4iT - 5.94e8T^{2}
31 1−5.33e3T+8.87e8T2 1 - 5.33e3T + 8.87e8T^{2}
37 1−3.25e4T+2.56e9T2 1 - 3.25e4T + 2.56e9T^{2}
41 1+6.61e4iT−4.75e9T2 1 + 6.61e4iT - 4.75e9T^{2}
43 1+7.06e4T+6.32e9T2 1 + 7.06e4T + 6.32e9T^{2}
47 1+3.98e3iT−1.07e10T2 1 + 3.98e3iT - 1.07e10T^{2}
53 1+1.90e5iT−2.21e10T2 1 + 1.90e5iT - 2.21e10T^{2}
59 1+2.37e5iT−4.21e10T2 1 + 2.37e5iT - 4.21e10T^{2}
61 1+6.18e4T+5.15e10T2 1 + 6.18e4T + 5.15e10T^{2}
67 1+4.30e5T+9.04e10T2 1 + 4.30e5T + 9.04e10T^{2}
71 1−2.51e5iT−1.28e11T2 1 - 2.51e5iT - 1.28e11T^{2}
73 1−2.51e5T+1.51e11T2 1 - 2.51e5T + 1.51e11T^{2}
79 1−6.60e5T+2.43e11T2 1 - 6.60e5T + 2.43e11T^{2}
83 1+7.97e5iT−3.26e11T2 1 + 7.97e5iT - 3.26e11T^{2}
89 1−2.70e5iT−4.96e11T2 1 - 2.70e5iT - 4.96e11T^{2}
97 1−2.20e5T+8.32e11T2 1 - 2.20e5T + 8.32e11T^{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

−15.67967349470792618529683367992, −14.83552838579428360049450296384, −13.57293222126820040829193728746, −11.62369540682971623391568410023, −11.05625979805148337460201676372, −9.990486139180967140208432314978, −7.60224623730087719735163842154, −6.36916925635655843099364052175, −3.53858300520273510886731839187, −2.03232292531174936133167360743, 1.37340758254179577886894900599, 4.68841056120705700074402565526, 6.00095655174430841648749994050, 8.111968754919570520732082179980, 8.733367916535631938827728697853, 11.01828444618274095997147346912, 12.29168464788704310873779048123, 13.60520871549574784535648973829, 15.05649146261494812633409485461, 16.32901971440025001174604379715

Graph of the ZZ-function along the critical line