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Properties

Label 2-3e6-27.4-c1-0-4
Degree $2$
Conductor $729$
Sign $-0.802 - 0.597i$
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  + (−1.85 − 0.673i)2-s + (1.43 + 1.20i)4-s + (−0.642 + 3.64i)5-s + (−1.79 + 1.50i)7-s + (0.118 + 0.205i)8-s + (3.64 − 6.31i)10-s + (0.378 + 2.14i)11-s + (4.43 − 1.61i)13-s + (4.34 − 1.58i)14-s + (−0.733 − 4.16i)16-s + (−1.46 + 2.54i)17-s + (3.11 + 5.39i)19-s + (−5.32 + 4.47i)20-s + (0.745 − 4.22i)22-s + (−0.397 − 0.333i)23-s + ⋯
L(s)  = 1  + (−1.30 − 0.476i)2-s + (0.719 + 0.604i)4-s + (−0.287 + 1.63i)5-s + (−0.679 + 0.570i)7-s + (0.0419 + 0.0727i)8-s + (1.15 − 1.99i)10-s + (0.114 + 0.646i)11-s + (1.22 − 0.447i)13-s + (1.16 − 0.422i)14-s + (−0.183 − 1.04i)16-s + (−0.355 + 0.616i)17-s + (0.714 + 1.23i)19-s + (−1.19 + 0.999i)20-s + (0.158 − 0.900i)22-s + (−0.0829 − 0.0695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.802 - 0.597i$
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: \((2,\ 729,\ (\ :1/2),\ -0.802 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139280 + 0.420323i\)
\(L(\frac12)\) \(\approx\) \(0.139280 + 0.420323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 0.673i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.642 - 3.64i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.79 - 1.50i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.378 - 2.14i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-4.43 + 1.61i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.46 - 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 5.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.397 + 0.333i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.28 + 1.19i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.29 + 2.76i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.34 + 0.854i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.184 - 1.04i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.181 + 0.152i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 + (-2.31 + 13.1i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.81 - 2.36i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (13.4 - 4.89i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.601 + 1.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.34 - 4.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.0 - 4.37i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (10.6 + 3.86i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.349 - 0.605i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.23 + 6.97i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.61701879536203007169243421526, −9.935048185876011715841622838099, −9.249706403702547839353833114583, −8.172849375139151043759602702134, −7.53248846063986408914228868227, −6.53623921474587373849393076359, −5.75005327870241179247453388393, −3.80186050372204447593869314525, −2.93225009246355363228501668370, −1.78278160196351368155497694082, 0.39157925542589686518637285951, 1.32651575353131340359931315930, 3.56394644721511209215636070520, 4.56688932750607381378120965244, 5.77953013232616860803247058955, 6.84145834298669143783990223786, 7.62284217410166765040013337741, 8.643451151275889276944172588992, 9.024193411210011766179354412828, 9.564310965217071656036677430047

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