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Properties

Label 2-3e3-27.2-c6-0-12
Degree 22
Conductor 2727
Sign −0.443+0.896i-0.443 + 0.896i
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  + (1.51 + 0.267i)2-s + (26.8 − 2.67i)3-s + (−57.9 − 21.0i)4-s + (−139. − 165. i)5-s + (41.4 + 3.13i)6-s + (−39.0 + 14.2i)7-s + (−167. − 96.6i)8-s + (714. − 143. i)9-s + (−166. − 288. i)10-s + (−170. + 203. i)11-s + (−1.61e3 − 411. i)12-s + (−454. − 2.57e3i)13-s + (−62.9 + 11.0i)14-s + (−4.18e3 − 4.08e3i)15-s + (2.79e3 + 2.34e3i)16-s + (4.58e3 − 2.64e3i)17-s + ⋯
L(s)  = 1  + (0.189 + 0.0333i)2-s + (0.995 − 0.0989i)3-s + (−0.904 − 0.329i)4-s + (−1.11 − 1.32i)5-s + (0.191 + 0.0144i)6-s + (−0.113 + 0.0414i)7-s + (−0.326 − 0.188i)8-s + (0.980 − 0.196i)9-s + (−0.166 − 0.288i)10-s + (−0.128 + 0.152i)11-s + (−0.933 − 0.238i)12-s + (−0.206 − 1.17i)13-s + (−0.0229 + 0.00404i)14-s + (−1.23 − 1.20i)15-s + (0.682 + 0.572i)16-s + (0.933 − 0.538i)17-s + ⋯

Functional equation

Λ(s)=(27s/2 ΓC(s) L(s)=((−0.443+0.896i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(27s/2 ΓC(s+3) L(s)=((−0.443+0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2727    =    333^{3}
Sign: −0.443+0.896i-0.443 + 0.896i
Analytic conductor: 6.211466.21146
Root analytic conductor: 2.492282.49228
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ27(2,⋅)\chi_{27} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 27, ( :3), −0.443+0.896i)(2,\ 27,\ (\ :3),\ -0.443 + 0.896i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.710165−1.14327i0.710165 - 1.14327i
L(12)L(\frac12) ≈\approx 0.710165−1.14327i0.710165 - 1.14327i
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+(−26.8+2.67i)T 1 + (-26.8 + 2.67i)T
good2 1+(−1.51−0.267i)T+(60.1+21.8i)T2 1 + (-1.51 - 0.267i)T + (60.1 + 21.8i)T^{2}
5 1+(139.+165.i)T+(−2.71e3+1.53e4i)T2 1 + (139. + 165. i)T + (-2.71e3 + 1.53e4i)T^{2}
7 1+(39.0−14.2i)T+(9.01e4−7.56e4i)T2 1 + (39.0 - 14.2i)T + (9.01e4 - 7.56e4i)T^{2}
11 1+(170.−203.i)T+(−3.07e5−1.74e6i)T2 1 + (170. - 203. i)T + (-3.07e5 - 1.74e6i)T^{2}
13 1+(454.+2.57e3i)T+(−4.53e6+1.65e6i)T2 1 + (454. + 2.57e3i)T + (-4.53e6 + 1.65e6i)T^{2}
17 1+(−4.58e3+2.64e3i)T+(1.20e7−2.09e7i)T2 1 + (-4.58e3 + 2.64e3i)T + (1.20e7 - 2.09e7i)T^{2}
19 1+(3.57e3−6.18e3i)T+(−2.35e7−4.07e7i)T2 1 + (3.57e3 - 6.18e3i)T + (-2.35e7 - 4.07e7i)T^{2}
23 1+(−4.13e3+1.13e4i)T+(−1.13e8−9.51e7i)T2 1 + (-4.13e3 + 1.13e4i)T + (-1.13e8 - 9.51e7i)T^{2}
29 1+(−1.49e4−2.63e3i)T+(5.58e8+2.03e8i)T2 1 + (-1.49e4 - 2.63e3i)T + (5.58e8 + 2.03e8i)T^{2}
31 1+(2.51e4+9.15e3i)T+(6.79e8+5.70e8i)T2 1 + (2.51e4 + 9.15e3i)T + (6.79e8 + 5.70e8i)T^{2}
37 1+(−7.67e3−1.32e4i)T+(−1.28e9+2.22e9i)T2 1 + (-7.67e3 - 1.32e4i)T + (-1.28e9 + 2.22e9i)T^{2}
41 1+(−9.75e4+1.71e4i)T+(4.46e9−1.62e9i)T2 1 + (-9.75e4 + 1.71e4i)T + (4.46e9 - 1.62e9i)T^{2}
43 1+(6.71e4+5.63e4i)T+(1.09e9+6.22e9i)T2 1 + (6.71e4 + 5.63e4i)T + (1.09e9 + 6.22e9i)T^{2}
47 1+(5.82e4+1.60e5i)T+(−8.25e9+6.92e9i)T2 1 + (5.82e4 + 1.60e5i)T + (-8.25e9 + 6.92e9i)T^{2}
53 1−8.80e3iT−2.21e10T2 1 - 8.80e3iT - 2.21e10T^{2}
59 1+(−1.07e5−1.27e5i)T+(−7.32e9+4.15e10i)T2 1 + (-1.07e5 - 1.27e5i)T + (-7.32e9 + 4.15e10i)T^{2}
61 1+(−4.06e5+1.48e5i)T+(3.94e10−3.31e10i)T2 1 + (-4.06e5 + 1.48e5i)T + (3.94e10 - 3.31e10i)T^{2}
67 1+(5.09e4+2.89e5i)T+(−8.50e10+3.09e10i)T2 1 + (5.09e4 + 2.89e5i)T + (-8.50e10 + 3.09e10i)T^{2}
71 1+(−1.92e5+1.10e5i)T+(6.40e10−1.10e11i)T2 1 + (-1.92e5 + 1.10e5i)T + (6.40e10 - 1.10e11i)T^{2}
73 1+(−1.48e5+2.57e5i)T+(−7.56e10−1.31e11i)T2 1 + (-1.48e5 + 2.57e5i)T + (-7.56e10 - 1.31e11i)T^{2}
79 1+(1.23e5−7.00e5i)T+(−2.28e11−8.31e10i)T2 1 + (1.23e5 - 7.00e5i)T + (-2.28e11 - 8.31e10i)T^{2}
83 1+(3.72e5+6.57e4i)T+(3.07e11+1.11e11i)T2 1 + (3.72e5 + 6.57e4i)T + (3.07e11 + 1.11e11i)T^{2}
89 1+(6.67e5+3.85e5i)T+(2.48e11+4.30e11i)T2 1 + (6.67e5 + 3.85e5i)T + (2.48e11 + 4.30e11i)T^{2}
97 1+(−5.76e5−4.83e5i)T+(1.44e11+8.20e11i)T2 1 + (-5.76e5 - 4.83e5i)T + (1.44e11 + 8.20e11i)T^{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

−15.41954137199437279658927075997, −14.44286984863694098856275619239, −12.94962242570940379912194899393, −12.42551964875352118015586905745, −9.986191376696520369354932486648, −8.678078485149980194173562674592, −7.87992864498215226642534739836, −5.05419508999541134645555264642, −3.71302458660262966545310862963, −0.69471934502924925791398832691, 3.08688909815106196958698049891, 4.19523087737649406128837474159, 7.11414802376640864643319296855, 8.274097680763505542163930910357, 9.703013013686356089879322028273, 11.32682388961255975742489832384, 12.86606509450028175902787902619, 14.24932729545923284471512537938, 14.79375580301738198757700441913, 16.12474837227987777327980111884

Graph of the ZZ-function along the critical line