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Properties

Label 2-3e6-243.103-c1-0-22
Degree 22
Conductor 729729
Sign 0.382+0.923i0.382 + 0.923i
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  + (1.39 + 0.387i)2-s + (0.0768 + 0.0463i)4-s + (−1.01 + 0.0395i)5-s + (−0.161 − 0.0252i)7-s + (−1.89 − 2.00i)8-s + (−1.43 − 0.340i)10-s + (−0.568 − 4.16i)11-s + (1.68 − 2.45i)13-s + (−0.215 − 0.0978i)14-s + (−1.94 − 3.69i)16-s + (6.87 − 3.45i)17-s + (0.307 + 5.27i)19-s + (−0.0802 − 0.0442i)20-s + (0.821 − 6.01i)22-s + (2.81 − 7.29i)23-s + ⋯
L(s)  = 1  + (0.984 + 0.274i)2-s + (0.0384 + 0.0231i)4-s + (−0.456 + 0.0176i)5-s + (−0.0611 − 0.00955i)7-s + (−0.669 − 0.710i)8-s + (−0.453 − 0.107i)10-s + (−0.171 − 1.25i)11-s + (0.467 − 0.682i)13-s + (−0.0575 − 0.0261i)14-s + (−0.486 − 0.922i)16-s + (1.66 − 0.837i)17-s + (0.0705 + 1.21i)19-s + (−0.0179 − 0.00989i)20-s + (0.175 − 1.28i)22-s + (0.587 − 1.52i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.382+0.923i0.382 + 0.923i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(685,⋅)\chi_{729} (685, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.382+0.923i)(2,\ 729,\ (\ :1/2),\ 0.382 + 0.923i)

Particular Values

L(1)L(1) ≈\approx 1.52152−1.01659i1.52152 - 1.01659i
L(12)L(\frac12) ≈\approx 1.52152−1.01659i1.52152 - 1.01659i
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+(−1.39−0.387i)T+(1.71+1.03i)T2 1 + (-1.39 - 0.387i)T + (1.71 + 1.03i)T^{2}
5 1+(1.01−0.0395i)T+(4.98−0.387i)T2 1 + (1.01 - 0.0395i)T + (4.98 - 0.387i)T^{2}
7 1+(0.161+0.0252i)T+(6.66+2.13i)T2 1 + (0.161 + 0.0252i)T + (6.66 + 2.13i)T^{2}
11 1+(0.568+4.16i)T+(−10.5+2.94i)T2 1 + (0.568 + 4.16i)T + (-10.5 + 2.94i)T^{2}
13 1+(−1.68+2.45i)T+(−4.68−12.1i)T2 1 + (-1.68 + 2.45i)T + (-4.68 - 12.1i)T^{2}
17 1+(−6.87+3.45i)T+(10.1−13.6i)T2 1 + (-6.87 + 3.45i)T + (10.1 - 13.6i)T^{2}
19 1+(−0.307−5.27i)T+(−18.8+2.20i)T2 1 + (-0.307 - 5.27i)T + (-18.8 + 2.20i)T^{2}
23 1+(−2.81+7.29i)T+(−17.0−15.4i)T2 1 + (-2.81 + 7.29i)T + (-17.0 - 15.4i)T^{2}
29 1+(7.18+5.13i)T+(9.38+27.4i)T2 1 + (7.18 + 5.13i)T + (9.38 + 27.4i)T^{2}
31 1+(−1.77+2.03i)T+(−4.19−30.7i)T2 1 + (-1.77 + 2.03i)T + (-4.19 - 30.7i)T^{2}
37 1+(−2.96−3.98i)T+(−10.6+35.4i)T2 1 + (-2.96 - 3.98i)T + (-10.6 + 35.4i)T^{2}
41 1+(2.73−10.6i)T+(−35.8−19.8i)T2 1 + (2.73 - 10.6i)T + (-35.8 - 19.8i)T^{2}
43 1+(−0.972−0.882i)T+(4.16+42.7i)T2 1 + (-0.972 - 0.882i)T + (4.16 + 42.7i)T^{2}
47 1+(−2.91−3.34i)T+(−6.36+46.5i)T2 1 + (-2.91 - 3.34i)T + (-6.36 + 46.5i)T^{2}
53 1+(−1.30+7.41i)T+(−49.8−18.1i)T2 1 + (-1.30 + 7.41i)T + (-49.8 - 18.1i)T^{2}
59 1+(9.14−3.73i)T+(42.1−41.3i)T2 1 + (9.14 - 3.73i)T + (42.1 - 41.3i)T^{2}
61 1+(1.32−0.801i)T+(28.4−53.9i)T2 1 + (1.32 - 0.801i)T + (28.4 - 53.9i)T^{2}
67 1+(−5.87+4.20i)T+(21.6−63.3i)T2 1 + (-5.87 + 4.20i)T + (21.6 - 63.3i)T^{2}
71 1+(−1.31−4.37i)T+(−59.3+39.0i)T2 1 + (-1.31 - 4.37i)T + (-59.3 + 39.0i)T^{2}
73 1+(−3.87+0.917i)T+(65.2−32.7i)T2 1 + (-3.87 + 0.917i)T + (65.2 - 32.7i)T^{2}
79 1+(−7.95+7.79i)T+(1.53−78.9i)T2 1 + (-7.95 + 7.79i)T + (1.53 - 78.9i)T^{2}
83 1+(1.44+5.60i)T+(−72.6+40.0i)T2 1 + (1.44 + 5.60i)T + (-72.6 + 40.0i)T^{2}
89 1+(1.18−3.94i)T+(−74.3−48.9i)T2 1 + (1.18 - 3.94i)T + (-74.3 - 48.9i)T^{2}
97 1+(−5.41−0.210i)T+(96.7+7.51i)T2 1 + (-5.41 - 0.210i)T + (96.7 + 7.51i)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

−10.19910913804457871739731669769, −9.487058809234202230725161627078, −8.233597689359697838193374363911, −7.72673872844008847060560377860, −6.25635289252315335435782096163, −5.79616263188837531912883929683, −4.82951488644132654615014055388, −3.65481000968587599641851496279, −3.08088886309032962873278489311, −0.71874771632796074597455271387, 1.85376810605837827297434089825, 3.33699177958311045636544308563, 3.99251382032091112328097746318, 5.03868904857739685652059526244, 5.76460891890493635958046350864, 7.10378057057733846017757743154, 7.79553169119261579375753691252, 8.974816958701404967053824421163, 9.623410104325637713886003450827, 10.84200833677158942803713159316

Graph of the ZZ-function along the critical line