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Properties

Label 2-3e3-3.2-c8-0-1
Degree 22
Conductor 2727
Sign 11
Analytic cond. 10.999210.9992
Root an. cond. 3.316503.31650
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.3i·2-s − 608·4-s + 823. i·5-s + 1.96e3·7-s + 1.03e4i·8-s + 2.41e4·10-s + 1.25e4i·11-s − 4.55e4·13-s − 5.78e4i·14-s + 1.48e5·16-s + 5.96e4i·17-s + 1.52e5·19-s − 5.00e5i·20-s + 3.69e5·22-s + 1.31e5i·23-s + ⋯
L(s)  = 1  − 1.83i·2-s − 2.37·4-s + 1.31i·5-s + 0.819·7-s + 2.52i·8-s + 2.41·10-s + 0.859i·11-s − 1.59·13-s − 1.50i·14-s + 2.26·16-s + 0.713i·17-s + 1.16·19-s − 3.12i·20-s + 1.57·22-s + 0.469i·23-s + ⋯

Functional equation

Λ(s)=(27s/2 ΓC(s) L(s)=( Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(27s/2 ΓC(s+4) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2727    =    333^{3}
Sign: 11
Analytic conductor: 10.999210.9992
Root analytic conductor: 3.316503.31650
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ27(26,⋅)\chi_{27} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 27, ( :4), 1)(2,\ 27,\ (\ :4),\ 1)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.106121.10612
L(12)L(\frac12) ≈\approx 1.106121.10612
L(5)L(5) 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+29.3iT−256T2 1 + 29.3iT - 256T^{2}
5 1−823.iT−3.90e5T2 1 - 823. iT - 3.90e5T^{2}
7 1−1.96e3T+5.76e6T2 1 - 1.96e3T + 5.76e6T^{2}
11 1−1.25e4iT−2.14e8T2 1 - 1.25e4iT - 2.14e8T^{2}
13 1+4.55e4T+8.15e8T2 1 + 4.55e4T + 8.15e8T^{2}
17 1−5.96e4iT−6.97e9T2 1 - 5.96e4iT - 6.97e9T^{2}
19 1−1.52e5T+1.69e10T2 1 - 1.52e5T + 1.69e10T^{2}
23 1−1.31e5iT−7.83e10T2 1 - 1.31e5iT - 7.83e10T^{2}
29 1−5.88e5iT−5.00e11T2 1 - 5.88e5iT - 5.00e11T^{2}
31 1+1.64e5T+8.52e11T2 1 + 1.64e5T + 8.52e11T^{2}
37 1+6.63e5T+3.51e12T2 1 + 6.63e5T + 3.51e12T^{2}
41 1−9.38e5iT−7.98e12T2 1 - 9.38e5iT - 7.98e12T^{2}
43 1−5.75e5T+1.16e13T2 1 - 5.75e5T + 1.16e13T^{2}
47 1−9.23e6iT−2.38e13T2 1 - 9.23e6iT - 2.38e13T^{2}
53 1+1.03e7iT−6.22e13T2 1 + 1.03e7iT - 6.22e13T^{2}
59 1−5.03e6iT−1.46e14T2 1 - 5.03e6iT - 1.46e14T^{2}
61 1+1.92e7T+1.91e14T2 1 + 1.92e7T + 1.91e14T^{2}
67 1+5.98e5T+4.06e14T2 1 + 5.98e5T + 4.06e14T^{2}
71 1+2.92e7iT−6.45e14T2 1 + 2.92e7iT - 6.45e14T^{2}
73 1−1.28e7T+8.06e14T2 1 - 1.28e7T + 8.06e14T^{2}
79 1+2.35e7T+1.51e15T2 1 + 2.35e7T + 1.51e15T^{2}
83 1+3.34e7iT−2.25e15T2 1 + 3.34e7iT - 2.25e15T^{2}
89 1−2.82e7iT−3.93e15T2 1 - 2.82e7iT - 3.93e15T^{2}
97 1−1.36e8T+7.83e15T2 1 - 1.36e8T + 7.83e15T^{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

−14.85796048591503592714676740965, −14.14734310171420113978962528196, −12.54063389911506352363865170605, −11.52604338076039343415075765899, −10.51158709903341614490408552913, −9.560412922687546050177047019007, −7.52063485810322983908469228758, −4.80754363752921529066023694918, −3.08754576563721815408088786126, −1.80398497337955035900751147504, 0.52040295694008958693358612029, 4.68732513555273015926947567644, 5.46432275902876930682036652135, 7.38257311318120953641060315951, 8.424502580573556382771707379568, 9.496711958628782159689328433733, 12.06162512665889783105698828169, 13.52445252718662654061100376604, 14.44884294482851358939031487818, 15.71400430337188836957098893216

Graph of the ZZ-function along the critical line