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Properties

Label 2-3e6-27.7-c1-0-31
Degree $2$
Conductor $729$
Sign $-0.686 + 0.727i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.62 − 0.592i)2-s + (0.766 − 0.642i)4-s + (−0.601 − 3.41i)5-s + (−0.766 − 0.642i)7-s + (−0.866 + 1.50i)8-s + (−3 − 5.19i)10-s + (0.601 − 3.41i)11-s + (−4.69 − 1.71i)13-s + (−1.62 − 0.592i)14-s + (−0.868 + 4.92i)16-s + (0.5 − 0.866i)19-s + (−2.65 − 2.22i)20-s + (−1.04 − 5.90i)22-s + (5.30 − 4.45i)23-s + (−6.57 + 2.39i)25-s − 8.66·26-s + ⋯
L(s)  = 1  + (1.15 − 0.418i)2-s + (0.383 − 0.321i)4-s + (−0.269 − 1.52i)5-s + (−0.289 − 0.242i)7-s + (−0.306 + 0.530i)8-s + (−0.948 − 1.64i)10-s + (0.181 − 1.02i)11-s + (−1.30 − 0.474i)13-s + (−0.434 − 0.158i)14-s + (−0.217 + 1.23i)16-s + (0.114 − 0.198i)19-s + (−0.593 − 0.497i)20-s + (−0.222 − 1.25i)22-s + (1.10 − 0.928i)23-s + (−1.31 + 0.478i)25-s − 1.69·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.686 + 0.727i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778844 - 1.80556i\)
\(L(\frac12)\) \(\approx\) \(0.778844 - 1.80556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-1.62 + 0.592i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.601 + 3.41i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.601 + 3.41i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (4.69 + 1.71i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.30 + 4.45i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.25 - 1.18i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.83 + 3.21i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.25 - 1.18i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.65 - 2.22i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (0.601 + 3.41i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.53 - 1.28i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (7.51 + 2.73i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.19 - 9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.51 + 2.36i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.95 + 16.7i)T + (-91.1 - 33.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.18253345473580018566523611149, −9.055849299523675695657542435022, −8.532180489083032133293326344913, −7.50949532042508603783597581093, −6.13228156309247196509911913567, −5.17318214293123050003173826987, −4.66183916712424468382060024720, −3.66115682801955215174962972470, −2.55624458821947458889557806415, −0.69099897615432776937132617573, 2.41784062240076514268593010533, 3.33284053236736739250900201017, 4.33639454917734883298579288395, 5.29511629789797108920924073454, 6.34004921234829549823961623344, 7.13932876488606960826246114824, 7.43778349231758602738597042726, 9.259946783447249186486706238237, 9.857041161763467967715189464685, 10.76413542160586636092466659752

Graph of the $Z$-function along the critical line