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Properties

Label 2-3e6-27.4-c1-0-21
Degree 22
Conductor 729729
Sign 0.893+0.448i0.893 + 0.448i
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  + (1.26 + 0.460i)2-s + (−0.141 − 0.118i)4-s + (−0.286 + 1.62i)5-s + (1.84 − 1.55i)7-s + (−1.47 − 2.54i)8-s + (−1.11 + 1.92i)10-s + (−1.03 − 5.85i)11-s + (3.03 − 1.10i)13-s + (3.05 − 1.11i)14-s + (−0.624 − 3.54i)16-s + (1.5 − 2.59i)17-s + (3.31 + 5.74i)19-s + (0.233 − 0.196i)20-s + (1.39 − 7.88i)22-s + (2.25 + 1.89i)23-s + ⋯
L(s)  = 1  + (0.895 + 0.325i)2-s + (−0.0707 − 0.0593i)4-s + (−0.128 + 0.727i)5-s + (0.698 − 0.585i)7-s + (−0.520 − 0.901i)8-s + (−0.352 + 0.609i)10-s + (−0.311 − 1.76i)11-s + (0.840 − 0.306i)13-s + (0.815 − 0.296i)14-s + (−0.156 − 0.885i)16-s + (0.363 − 0.630i)17-s + (0.761 + 1.31i)19-s + (0.0523 − 0.0438i)20-s + (0.296 − 1.68i)22-s + (0.470 + 0.394i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.893+0.448i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.893+0.448i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.893+0.448i0.893 + 0.448i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(568,⋅)\chi_{729} (568, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.893+0.448i)(2,\ 729,\ (\ :1/2),\ 0.893 + 0.448i)

Particular Values

L(1)L(1) ≈\approx 2.20050−0.521530i2.20050 - 0.521530i
L(12)L(\frac12) ≈\approx 2.20050−0.521530i2.20050 - 0.521530i
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+(−1.26−0.460i)T+(1.53+1.28i)T2 1 + (-1.26 - 0.460i)T + (1.53 + 1.28i)T^{2}
5 1+(0.286−1.62i)T+(−4.69−1.71i)T2 1 + (0.286 - 1.62i)T + (-4.69 - 1.71i)T^{2}
7 1+(−1.84+1.55i)T+(1.21−6.89i)T2 1 + (-1.84 + 1.55i)T + (1.21 - 6.89i)T^{2}
11 1+(1.03+5.85i)T+(−10.3+3.76i)T2 1 + (1.03 + 5.85i)T + (-10.3 + 3.76i)T^{2}
13 1+(−3.03+1.10i)T+(9.95−8.35i)T2 1 + (-3.03 + 1.10i)T + (9.95 - 8.35i)T^{2}
17 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.31−5.74i)T+(−9.5+16.4i)T2 1 + (-3.31 - 5.74i)T + (-9.5 + 16.4i)T^{2}
23 1+(−2.25−1.89i)T+(3.99+22.6i)T2 1 + (-2.25 - 1.89i)T + (3.99 + 22.6i)T^{2}
29 1+(1.21+0.441i)T+(22.2+18.6i)T2 1 + (1.21 + 0.441i)T + (22.2 + 18.6i)T^{2}
31 1+(0.450+0.378i)T+(5.38+30.5i)T2 1 + (0.450 + 0.378i)T + (5.38 + 30.5i)T^{2}
37 1+(0.0209−0.0362i)T+(−18.5−32.0i)T2 1 + (0.0209 - 0.0362i)T + (-18.5 - 32.0i)T^{2}
41 1+(4.60−1.67i)T+(31.4−26.3i)T2 1 + (4.60 - 1.67i)T + (31.4 - 26.3i)T^{2}
43 1+(0.900+5.10i)T+(−40.4+14.7i)T2 1 + (0.900 + 5.10i)T + (-40.4 + 14.7i)T^{2}
47 1+(−2.86+2.40i)T+(8.16−46.2i)T2 1 + (-2.86 + 2.40i)T + (8.16 - 46.2i)T^{2}
53 1+11.6T+53T2 1 + 11.6T + 53T^{2}
59 1+(−1.27+7.23i)T+(−55.4−20.1i)T2 1 + (-1.27 + 7.23i)T + (-55.4 - 20.1i)T^{2}
61 1+(−8.46+7.10i)T+(10.5−60.0i)T2 1 + (-8.46 + 7.10i)T + (10.5 - 60.0i)T^{2}
67 1+(1.74−0.635i)T+(51.3−43.0i)T2 1 + (1.74 - 0.635i)T + (51.3 - 43.0i)T^{2}
71 1+(2.75−4.77i)T+(−35.5−61.4i)T2 1 + (2.75 - 4.77i)T + (-35.5 - 61.4i)T^{2}
73 1+(2.77+4.81i)T+(−36.5+63.2i)T2 1 + (2.77 + 4.81i)T + (-36.5 + 63.2i)T^{2}
79 1+(−3.55−1.29i)T+(60.5+50.7i)T2 1 + (-3.55 - 1.29i)T + (60.5 + 50.7i)T^{2}
83 1+(−3.74−1.36i)T+(63.5+53.3i)T2 1 + (-3.74 - 1.36i)T + (63.5 + 53.3i)T^{2}
89 1+(−4.07−7.05i)T+(−44.5+77.0i)T2 1 + (-4.07 - 7.05i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.0452−0.256i)T+(−91.1+33.1i)T2 1 + (-0.0452 - 0.256i)T + (-91.1 + 33.1i)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

−10.58211041567997647939209598285, −9.544076401918778185432439829415, −8.415551952473219446843119921299, −7.64030319586250636816337596084, −6.59301207249055632953795093993, −5.73002739886537774799115562035, −5.07172410766409037885572274729, −3.63018272960433066693562651135, −3.27037687333684196147575573079, −1.00415111810951139295098173432, 1.68475625065057087344874778634, 2.89814604064344455828142901284, 4.29422544905455761499982613800, 4.82526419880881362444051952003, 5.54361729655722130458877194582, 6.89911928015032437836930091210, 8.010160754765619624216276946951, 8.757503974726403944138482369775, 9.441230571549330811705717523776, 10.70111505598932622237773454447

Graph of the ZZ-function along the critical line