Properties

Label 2-3e6-27.4-c1-0-16
Degree $2$
Conductor $729$
Sign $-0.0581 - 0.998i$
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  + (2.37 + 0.866i)2-s + (3.37 + 2.83i)4-s + (−0.0812 + 0.460i)5-s + (−2.47 + 2.07i)7-s + (3.05 + 5.28i)8-s + (−0.592 + 1.02i)10-s + (0.539 + 3.05i)11-s + (2.05 − 0.747i)13-s + (−7.67 + 2.79i)14-s + (1.15 + 6.53i)16-s + (1.5 − 2.59i)17-s + (−0.0209 − 0.0362i)19-s + (−1.58 + 1.32i)20-s + (−1.36 + 7.74i)22-s + (−4.67 − 3.92i)23-s + ⋯
L(s)  = 1  + (1.68 + 0.612i)2-s + (1.68 + 1.41i)4-s + (−0.0363 + 0.206i)5-s + (−0.934 + 0.783i)7-s + (1.07 + 1.86i)8-s + (−0.187 + 0.324i)10-s + (0.162 + 0.922i)11-s + (0.569 − 0.207i)13-s + (−2.05 + 0.746i)14-s + (0.288 + 1.63i)16-s + (0.363 − 0.630i)17-s + (−0.00480 − 0.00832i)19-s + (−0.353 + 0.296i)20-s + (−0.291 + 1.65i)22-s + (−0.975 − 0.818i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.0581 - 0.998i$
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: \((2,\ 729,\ (\ :1/2),\ -0.0581 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52166 + 2.67280i\)
\(L(\frac12)\) \(\approx\) \(2.52166 + 2.67280i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 - 0.866i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.0812 - 0.460i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (2.47 - 2.07i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.539 - 3.05i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-2.05 + 0.747i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0209 + 0.0362i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.67 + 3.92i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-6.17 - 2.24i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.76 + 4.00i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.79 - 3.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.23 + 2.63i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.102 + 0.579i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.40 + 6.20i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 + (-1.48 + 8.40i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.971 - 0.815i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (9.40 - 3.42i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-5.91 + 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.11 - 7.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.3 + 3.77i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.41 + 0.516i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (7.93 + 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.23 - 18.3i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.85940383750285413675460280550, −9.802119571378896860390402737052, −8.768336618569824948477490675895, −7.58498897922067043656716548860, −6.76208138739877573405766192770, −6.11474882552887929582592192987, −5.26912491901641081810838465170, −4.28469084225015372729719388170, −3.26996753321113511141156833973, −2.40615078735739029379176355235, 1.24079584972774686738362277000, 2.88251453772076108895226351368, 3.70073926471724745582060194903, 4.36025548447866173911538772970, 5.69609799201081114586181535754, 6.21016745354160102399001971201, 7.14476928520651747175142878539, 8.452274416383808175080923055614, 9.655670388346522389186077445917, 10.60993484941809614282451440358

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