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Properties

Label 2-3e6-27.16-c1-0-8
Degree 22
Conductor 729729
Sign 0.973+0.230i0.973 + 0.230i
Analytic cond. 5.821095.82109
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 1.11i)2-s + (0.173 + 0.984i)4-s + (3.25 + 1.18i)5-s + (−0.173 + 0.984i)7-s + (−0.866 + 1.5i)8-s + (−2.99 − 5.19i)10-s + (−3.25 + 1.18i)11-s + (3.83 − 3.21i)13-s + (1.32 − 1.11i)14-s + (4.69 − 1.71i)16-s + (0.5 − 0.866i)19-s + (−0.601 + 3.41i)20-s + (5.63 + 2.05i)22-s + (1.20 + 6.82i)23-s + (5.36 + 4.49i)25-s − 8.66·26-s + ⋯
L(s)  = 1  + (−0.938 − 0.787i)2-s + (0.0868 + 0.492i)4-s + (1.45 + 0.529i)5-s + (−0.0656 + 0.372i)7-s + (−0.306 + 0.530i)8-s + (−0.948 − 1.64i)10-s + (−0.981 + 0.357i)11-s + (1.06 − 0.891i)13-s + (0.354 − 0.297i)14-s + (1.17 − 0.427i)16-s + (0.114 − 0.198i)19-s + (−0.134 + 0.762i)20-s + (1.20 + 0.437i)22-s + (0.250 + 1.42i)23-s + (1.07 + 0.899i)25-s − 1.69·26-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.973+0.230i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.973+0.230i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.973+0.230i0.973 + 0.230i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(406,⋅)\chi_{729} (406, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.973+0.230i)(2,\ 729,\ (\ :1/2),\ 0.973 + 0.230i)

Particular Values

L(1)L(1) ≈\approx 1.12452−0.131437i1.12452 - 0.131437i
L(12)L(\frac12) ≈\approx 1.12452−0.131437i1.12452 - 0.131437i
L(32)L(\frac{3}{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)
bad3 1 1
good2 1+(1.32+1.11i)T+(0.347+1.96i)T2 1 + (1.32 + 1.11i)T + (0.347 + 1.96i)T^{2}
5 1+(−3.25−1.18i)T+(3.83+3.21i)T2 1 + (-3.25 - 1.18i)T + (3.83 + 3.21i)T^{2}
7 1+(0.173−0.984i)T+(−6.57−2.39i)T2 1 + (0.173 - 0.984i)T + (-6.57 - 2.39i)T^{2}
11 1+(3.25−1.18i)T+(8.42−7.07i)T2 1 + (3.25 - 1.18i)T + (8.42 - 7.07i)T^{2}
13 1+(−3.83+3.21i)T+(2.25−12.8i)T2 1 + (-3.83 + 3.21i)T + (2.25 - 12.8i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(−0.5+0.866i)T+(−9.5−16.4i)T2 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.20−6.82i)T+(−21.6+7.86i)T2 1 + (-1.20 - 6.82i)T + (-21.6 + 7.86i)T^{2}
29 1+(−2.65−2.22i)T+(5.03+28.5i)T2 1 + (-2.65 - 2.22i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.868−4.92i)T+(−29.1+10.6i)T2 1 + (-0.868 - 4.92i)T + (-29.1 + 10.6i)T^{2}
37 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.65−2.22i)T+(7.11−40.3i)T2 1 + (2.65 - 2.22i)T + (7.11 - 40.3i)T^{2}
43 1+(−0.939+0.342i)T+(32.9−27.6i)T2 1 + (-0.939 + 0.342i)T + (32.9 - 27.6i)T^{2}
47 1+(−0.601+3.41i)T+(−44.1−16.0i)T2 1 + (-0.601 + 3.41i)T + (-44.1 - 16.0i)T^{2}
53 1−10.3T+53T2 1 - 10.3T + 53T^{2}
59 1+(−3.25−1.18i)T+(45.1+37.9i)T2 1 + (-3.25 - 1.18i)T + (45.1 + 37.9i)T^{2}
61 1+(−0.347+1.96i)T+(−57.3−20.8i)T2 1 + (-0.347 + 1.96i)T + (-57.3 - 20.8i)T^{2}
67 1+(−6.12+5.14i)T+(11.6−65.9i)T2 1 + (-6.12 + 5.14i)T + (11.6 - 65.9i)T^{2}
71 1+(−5.19−9i)T+(−35.5+61.4i)T2 1 + (-5.19 - 9i)T + (-35.5 + 61.4i)T^{2}
73 1+(1−1.73i)T+(−36.5−63.2i)T2 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.766+0.642i)T+(13.7+77.7i)T2 1 + (0.766 + 0.642i)T + (13.7 + 77.7i)T^{2}
83 1+(5.30+4.45i)T+(14.4+81.7i)T2 1 + (5.30 + 4.45i)T + (14.4 + 81.7i)T^{2}
89 1+(−5.19+9i)T+(−44.5−77.0i)T2 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2}
97 1+(15.9−5.81i)T+(74.3−62.3i)T2 1 + (15.9 - 5.81i)T + (74.3 - 62.3i)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

−10.33482212270547605630066930510, −9.716318304400094777650370104285, −8.907419881708049467630124830791, −8.120916185345174593375360539903, −6.91443365739882641332494350958, −5.69145901754608142307092514204, −5.35679754117223076923575777301, −3.19886469357307466710158690200, −2.41283335485612933447174831786, −1.31639878977277672506223145894, 0.921340825796910299897261123653, 2.42420870351103210421677174294, 4.05666684278221958750301188997, 5.40988075636041190519329002579, 6.19684865338973261968586990953, 6.86315694199975006540617092476, 8.069815103481042698855808048060, 8.694374034091340172295435063525, 9.389769121730404211990219652602, 10.16981892718208645872715772854

Graph of the ZZ-function along the critical line