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Properties

Label 2-3e3-27.2-c6-0-4
Degree 22
Conductor 2727
Sign −0.226−0.973i-0.226 - 0.973i
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  + (−0.698 − 0.123i)2-s + (20.3 + 17.7i)3-s + (−59.6 − 21.7i)4-s + (95.6 + 113. i)5-s + (−12.0 − 14.9i)6-s + (−285. + 103. i)7-s + (78.3 + 45.2i)8-s + (96.7 + 722. i)9-s + (−52.7 − 91.3i)10-s + (−448. + 534. i)11-s + (−826. − 1.50e3i)12-s + (509. + 2.88e3i)13-s + (212. − 37.4i)14-s + (−83.1 + 4.01e3i)15-s + (3.06e3 + 2.57e3i)16-s + (−3.11e3 + 1.79e3i)17-s + ⋯
L(s)  = 1  + (−0.0873 − 0.0153i)2-s + (0.752 + 0.658i)3-s + (−0.932 − 0.339i)4-s + (0.764 + 0.911i)5-s + (−0.0555 − 0.0690i)6-s + (−0.832 + 0.303i)7-s + (0.152 + 0.0883i)8-s + (0.132 + 0.991i)9-s + (−0.0527 − 0.0913i)10-s + (−0.336 + 0.401i)11-s + (−0.478 − 0.869i)12-s + (0.231 + 1.31i)13-s + (0.0773 − 0.0136i)14-s + (−0.0246 + 1.18i)15-s + (0.748 + 0.627i)16-s + (−0.633 + 0.365i)17-s + ⋯

Functional equation

Λ(s)=(27s/2 ΓC(s) L(s)=((−0.226−0.973i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(27s/2 ΓC(s+3) L(s)=((−0.226−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2727    =    333^{3}
Sign: −0.226−0.973i-0.226 - 0.973i
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), −0.226−0.973i)(2,\ 27,\ (\ :3),\ -0.226 - 0.973i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.918777+1.15729i0.918777 + 1.15729i
L(12)L(\frac12) ≈\approx 0.918777+1.15729i0.918777 + 1.15729i
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+(−20.3−17.7i)T 1 + (-20.3 - 17.7i)T
good2 1+(0.698+0.123i)T+(60.1+21.8i)T2 1 + (0.698 + 0.123i)T + (60.1 + 21.8i)T^{2}
5 1+(−95.6−113.i)T+(−2.71e3+1.53e4i)T2 1 + (-95.6 - 113. i)T + (-2.71e3 + 1.53e4i)T^{2}
7 1+(285.−103.i)T+(9.01e4−7.56e4i)T2 1 + (285. - 103. i)T + (9.01e4 - 7.56e4i)T^{2}
11 1+(448.−534.i)T+(−3.07e5−1.74e6i)T2 1 + (448. - 534. i)T + (-3.07e5 - 1.74e6i)T^{2}
13 1+(−509.−2.88e3i)T+(−4.53e6+1.65e6i)T2 1 + (-509. - 2.88e3i)T + (-4.53e6 + 1.65e6i)T^{2}
17 1+(3.11e3−1.79e3i)T+(1.20e7−2.09e7i)T2 1 + (3.11e3 - 1.79e3i)T + (1.20e7 - 2.09e7i)T^{2}
19 1+(−4.61e3+7.99e3i)T+(−2.35e7−4.07e7i)T2 1 + (-4.61e3 + 7.99e3i)T + (-2.35e7 - 4.07e7i)T^{2}
23 1+(−7.85e3+2.15e4i)T+(−1.13e8−9.51e7i)T2 1 + (-7.85e3 + 2.15e4i)T + (-1.13e8 - 9.51e7i)T^{2}
29 1+(−1.94e4−3.42e3i)T+(5.58e8+2.03e8i)T2 1 + (-1.94e4 - 3.42e3i)T + (5.58e8 + 2.03e8i)T^{2}
31 1+(−4.80e3−1.74e3i)T+(6.79e8+5.70e8i)T2 1 + (-4.80e3 - 1.74e3i)T + (6.79e8 + 5.70e8i)T^{2}
37 1+(−7.32e3−1.26e4i)T+(−1.28e9+2.22e9i)T2 1 + (-7.32e3 - 1.26e4i)T + (-1.28e9 + 2.22e9i)T^{2}
41 1+(1.90e4−3.35e3i)T+(4.46e9−1.62e9i)T2 1 + (1.90e4 - 3.35e3i)T + (4.46e9 - 1.62e9i)T^{2}
43 1+(−1.10e5−9.29e4i)T+(1.09e9+6.22e9i)T2 1 + (-1.10e5 - 9.29e4i)T + (1.09e9 + 6.22e9i)T^{2}
47 1+(1.39e3+3.83e3i)T+(−8.25e9+6.92e9i)T2 1 + (1.39e3 + 3.83e3i)T + (-8.25e9 + 6.92e9i)T^{2}
53 1+1.80e5iT−2.21e10T2 1 + 1.80e5iT - 2.21e10T^{2}
59 1+(−1.73e5−2.07e5i)T+(−7.32e9+4.15e10i)T2 1 + (-1.73e5 - 2.07e5i)T + (-7.32e9 + 4.15e10i)T^{2}
61 1+(1.29e5−4.69e4i)T+(3.94e10−3.31e10i)T2 1 + (1.29e5 - 4.69e4i)T + (3.94e10 - 3.31e10i)T^{2}
67 1+(−1.02e4−5.78e4i)T+(−8.50e10+3.09e10i)T2 1 + (-1.02e4 - 5.78e4i)T + (-8.50e10 + 3.09e10i)T^{2}
71 1+(2.05e5−1.18e5i)T+(6.40e10−1.10e11i)T2 1 + (2.05e5 - 1.18e5i)T + (6.40e10 - 1.10e11i)T^{2}
73 1+(−2.11e5+3.65e5i)T+(−7.56e10−1.31e11i)T2 1 + (-2.11e5 + 3.65e5i)T + (-7.56e10 - 1.31e11i)T^{2}
79 1+(3.28e4−1.86e5i)T+(−2.28e11−8.31e10i)T2 1 + (3.28e4 - 1.86e5i)T + (-2.28e11 - 8.31e10i)T^{2}
83 1+(−7.57e5−1.33e5i)T+(3.07e11+1.11e11i)T2 1 + (-7.57e5 - 1.33e5i)T + (3.07e11 + 1.11e11i)T^{2}
89 1+(8.35e5+4.82e5i)T+(2.48e11+4.30e11i)T2 1 + (8.35e5 + 4.82e5i)T + (2.48e11 + 4.30e11i)T^{2}
97 1+(−1.19e6−1.00e6i)T+(1.44e11+8.20e11i)T2 1 + (-1.19e6 - 1.00e6i)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

−16.22304980298092642850845391862, −14.89769115461049387050207912845, −14.02196893423998630971242165336, −13.09563773321623590058645603271, −10.70023779045738467912301462931, −9.677867596498864548615305212779, −8.828446062978237835007140268756, −6.57211111343127118763223542853, −4.57764854979561784574286590662, −2.63593367198022911587396569889, 0.857538616887083412294377880335, 3.37142174333377133299333390633, 5.58880023407254327229130450129, 7.71642963415135658488187489319, 8.946958612509028558319028488376, 9.877702747018239101118963941912, 12.49794629353305759483766016879, 13.28607467734570921387385939021, 13.84967936176226846065502526685, 15.73352296927837285235346848928

Graph of the ZZ-function along the critical line