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Properties

Label 2-3e6-27.25-c1-0-4
Degree 22
Conductor 729729
Sign −0.686+0.727i-0.686 + 0.727i
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  + (−0.300 + 1.70i)2-s + (−0.939 − 0.342i)4-s + (−2.65 + 2.22i)5-s + (0.939 − 0.342i)7-s + (−0.866 + 1.50i)8-s + (−2.99 − 5.19i)10-s + (2.65 + 2.22i)11-s + (0.868 + 4.92i)13-s + (0.300 + 1.70i)14-s + (−3.83 − 3.21i)16-s + (0.5 − 0.866i)19-s + (3.25 − 1.18i)20-s + (−4.59 + 3.85i)22-s + (−6.51 − 2.36i)23-s + (1.21 − 6.89i)25-s − 8.66·26-s + ⋯
L(s)  = 1  + (−0.212 + 1.20i)2-s + (−0.469 − 0.171i)4-s + (−1.18 + 0.995i)5-s + (0.355 − 0.129i)7-s + (−0.306 + 0.530i)8-s + (−0.948 − 1.64i)10-s + (0.800 + 0.671i)11-s + (0.240 + 1.36i)13-s + (0.0803 + 0.455i)14-s + (−0.957 − 0.803i)16-s + (0.114 − 0.198i)19-s + (0.727 − 0.264i)20-s + (−0.979 + 0.822i)22-s + (−1.35 − 0.494i)23-s + (0.243 − 1.37i)25-s − 1.69·26-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((−0.686+0.727i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((−0.686+0.727i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: −0.686+0.727i-0.686 + 0.727i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(649,⋅)\chi_{729} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), −0.686+0.727i)(2,\ 729,\ (\ :1/2),\ -0.686 + 0.727i)

Particular Values

L(1)L(1) ≈\approx 0.330411−0.765978i0.330411 - 0.765978i
L(12)L(\frac12) ≈\approx 0.330411−0.765978i0.330411 - 0.765978i
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+(0.300−1.70i)T+(−1.87−0.684i)T2 1 + (0.300 - 1.70i)T + (-1.87 - 0.684i)T^{2}
5 1+(2.65−2.22i)T+(0.868−4.92i)T2 1 + (2.65 - 2.22i)T + (0.868 - 4.92i)T^{2}
7 1+(−0.939+0.342i)T+(5.36−4.49i)T2 1 + (-0.939 + 0.342i)T + (5.36 - 4.49i)T^{2}
11 1+(−2.65−2.22i)T+(1.91+10.8i)T2 1 + (-2.65 - 2.22i)T + (1.91 + 10.8i)T^{2}
13 1+(−0.868−4.92i)T+(−12.2+4.44i)T2 1 + (-0.868 - 4.92i)T + (-12.2 + 4.44i)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+(6.51+2.36i)T+(17.6+14.7i)T2 1 + (6.51 + 2.36i)T + (17.6 + 14.7i)T^{2}
29 1+(−0.601+3.41i)T+(−27.2−9.91i)T2 1 + (-0.601 + 3.41i)T + (-27.2 - 9.91i)T^{2}
31 1+(4.69+1.71i)T+(23.7+19.9i)T2 1 + (4.69 + 1.71i)T + (23.7 + 19.9i)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+(0.601+3.41i)T+(−38.5+14.0i)T2 1 + (0.601 + 3.41i)T + (-38.5 + 14.0i)T^{2}
43 1+(0.766+0.642i)T+(7.46+42.3i)T2 1 + (0.766 + 0.642i)T + (7.46 + 42.3i)T^{2}
47 1+(3.25−1.18i)T+(36.0−30.2i)T2 1 + (3.25 - 1.18i)T + (36.0 - 30.2i)T^{2}
53 1−10.3T+53T2 1 - 10.3T + 53T^{2}
59 1+(2.65−2.22i)T+(10.2−58.1i)T2 1 + (2.65 - 2.22i)T + (10.2 - 58.1i)T^{2}
61 1+(1.87−0.684i)T+(46.7−39.2i)T2 1 + (1.87 - 0.684i)T + (46.7 - 39.2i)T^{2}
67 1+(−1.38−7.87i)T+(−62.9+22.9i)T2 1 + (-1.38 - 7.87i)T + (-62.9 + 22.9i)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+(0.173−0.984i)T+(−74.2−27.0i)T2 1 + (0.173 - 0.984i)T + (-74.2 - 27.0i)T^{2}
83 1+(1.20−6.82i)T+(−77.9−28.3i)T2 1 + (1.20 - 6.82i)T + (-77.9 - 28.3i)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+(−13.0−10.9i)T+(16.8+95.5i)T2 1 + (-13.0 - 10.9i)T + (16.8 + 95.5i)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

−11.14629556924025272178272659423, −9.923268502601765814856170456217, −8.906914236716621049927362324643, −8.088164994621008835861132147430, −7.29003073623305743116255231135, −6.82689875289288719410760460769, −6.01793838319561445440966747196, −4.52336858015314157221092221211, −3.80824384368043512091731142441, −2.21918717256222236748963426204, 0.46999858994274609726907188939, 1.60532324430562213869251787458, 3.31501171786304366492844938198, 3.86222157574954521277513250126, 5.06476211853572887100472274781, 6.18872168982613289868357670276, 7.60349128906222541354859185430, 8.360190269817817939841212416721, 9.009199808918370131259393133197, 10.00651766546700391743742269593

Graph of the ZZ-function along the critical line