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Properties

Label 2-3e6-27.7-c1-0-14
Degree 22
Conductor 729729
Sign 0.802−0.597i0.802 - 0.597i
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  + (1.85 − 0.673i)2-s + (1.43 − 1.20i)4-s + (0.642 + 3.64i)5-s + (−1.79 − 1.50i)7-s + (−0.118 + 0.205i)8-s + (3.64 + 6.31i)10-s + (−0.378 + 2.14i)11-s + (4.43 + 1.61i)13-s + (−4.34 − 1.58i)14-s + (−0.733 + 4.16i)16-s + (1.46 + 2.54i)17-s + (3.11 − 5.39i)19-s + (5.32 + 4.47i)20-s + (0.745 + 4.22i)22-s + (0.397 − 0.333i)23-s + ⋯
L(s)  = 1  + (1.30 − 0.476i)2-s + (0.719 − 0.604i)4-s + (0.287 + 1.63i)5-s + (−0.679 − 0.570i)7-s + (−0.0419 + 0.0727i)8-s + (1.15 + 1.99i)10-s + (−0.114 + 0.646i)11-s + (1.22 + 0.447i)13-s + (−1.16 − 0.422i)14-s + (−0.183 + 1.04i)16-s + (0.355 + 0.616i)17-s + (0.714 − 1.23i)19-s + (1.19 + 0.999i)20-s + (0.158 + 0.900i)22-s + (0.0829 − 0.0695i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.802−0.597i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.802−0.597i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.802−0.597i0.802 - 0.597i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(163,⋅)\chi_{729} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.802−0.597i)(2,\ 729,\ (\ :1/2),\ 0.802 - 0.597i)

Particular Values

L(1)L(1) ≈\approx 2.75323+0.912321i2.75323 + 0.912321i
L(12)L(\frac12) ≈\approx 2.75323+0.912321i2.75323 + 0.912321i
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+(−1.85+0.673i)T+(1.53−1.28i)T2 1 + (-1.85 + 0.673i)T + (1.53 - 1.28i)T^{2}
5 1+(−0.642−3.64i)T+(−4.69+1.71i)T2 1 + (-0.642 - 3.64i)T + (-4.69 + 1.71i)T^{2}
7 1+(1.79+1.50i)T+(1.21+6.89i)T2 1 + (1.79 + 1.50i)T + (1.21 + 6.89i)T^{2}
11 1+(0.378−2.14i)T+(−10.3−3.76i)T2 1 + (0.378 - 2.14i)T + (-10.3 - 3.76i)T^{2}
13 1+(−4.43−1.61i)T+(9.95+8.35i)T2 1 + (-4.43 - 1.61i)T + (9.95 + 8.35i)T^{2}
17 1+(−1.46−2.54i)T+(−8.5+14.7i)T2 1 + (-1.46 - 2.54i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.11+5.39i)T+(−9.5−16.4i)T2 1 + (-3.11 + 5.39i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.397+0.333i)T+(3.99−22.6i)T2 1 + (-0.397 + 0.333i)T + (3.99 - 22.6i)T^{2}
29 1+(−3.28+1.19i)T+(22.2−18.6i)T2 1 + (-3.28 + 1.19i)T + (22.2 - 18.6i)T^{2}
31 1+(3.29−2.76i)T+(5.38−30.5i)T2 1 + (3.29 - 2.76i)T + (5.38 - 30.5i)T^{2}
37 1+(1.20+2.08i)T+(−18.5+32.0i)T2 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.34+0.854i)T+(31.4+26.3i)T2 1 + (2.34 + 0.854i)T + (31.4 + 26.3i)T^{2}
43 1+(−0.184+1.04i)T+(−40.4−14.7i)T2 1 + (-0.184 + 1.04i)T + (-40.4 - 14.7i)T^{2}
47 1+(0.181+0.152i)T+(8.16+46.2i)T2 1 + (0.181 + 0.152i)T + (8.16 + 46.2i)T^{2}
53 1−4.66T+53T2 1 - 4.66T + 53T^{2}
59 1+(2.31+13.1i)T+(−55.4+20.1i)T2 1 + (2.31 + 13.1i)T + (-55.4 + 20.1i)T^{2}
61 1+(2.81+2.36i)T+(10.5+60.0i)T2 1 + (2.81 + 2.36i)T + (10.5 + 60.0i)T^{2}
67 1+(13.4+4.89i)T+(51.3+43.0i)T2 1 + (13.4 + 4.89i)T + (51.3 + 43.0i)T^{2}
71 1+(0.601+1.04i)T+(−35.5+61.4i)T2 1 + (0.601 + 1.04i)T + (-35.5 + 61.4i)T^{2}
73 1+(−2.34+4.05i)T+(−36.5−63.2i)T2 1 + (-2.34 + 4.05i)T + (-36.5 - 63.2i)T^{2}
79 1+(−12.0+4.37i)T+(60.5−50.7i)T2 1 + (-12.0 + 4.37i)T + (60.5 - 50.7i)T^{2}
83 1+(−10.6+3.86i)T+(63.5−53.3i)T2 1 + (-10.6 + 3.86i)T + (63.5 - 53.3i)T^{2}
89 1+(0.349−0.605i)T+(−44.5−77.0i)T2 1 + (0.349 - 0.605i)T + (-44.5 - 77.0i)T^{2}
97 1+(1.23−6.97i)T+(−91.1−33.1i)T2 1 + (1.23 - 6.97i)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

−10.73901511448606785113709791480, −10.03959652839136344048027910115, −8.928675092189099830093709040861, −7.48722778393807856708695883619, −6.62177940758671867495144187830, −6.13903941145718199610070006540, −4.89085619756128327786629267042, −3.65081923333500062862001460880, −3.23049362361987352917731275373, −2.05309941515337270880963107204, 1.07841329206853991397378572291, 3.06847818781508330570251347320, 3.96121184183489133818384992833, 5.08971045413786372665019766930, 5.72747501130122605917182351410, 6.18606099916123878160368494591, 7.64318669703217292689142543088, 8.634843479952912711928818461358, 9.268898022003988219324914036369, 10.19123643079918405126108429171

Graph of the ZZ-function along the critical line