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Properties

Label 2-3e6-27.4-c1-0-6
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: $\chi_{729} (568, \cdot )$
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

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

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