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Properties

Label 2-3e6-27.25-c1-0-21
Degree $2$
Conductor $729$
Sign $0.973 + 0.230i$
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  + (−0.425 + 2.41i)2-s + (−3.75 − 1.36i)4-s + (1.87 − 1.57i)5-s + (−1.87 + 0.684i)7-s + (2.44 − 4.24i)8-s + (3.00 + 5.19i)10-s + (−1.87 − 1.57i)11-s + (−0.173 − 0.984i)13-s + (−0.850 − 4.82i)14-s + (3.06 + 2.57i)16-s + (−3.67 − 6.36i)17-s + (0.5 − 0.866i)19-s + (−9.20 + 3.35i)20-s + (4.59 − 3.85i)22-s + (−2.30 − 0.837i)23-s + ⋯
L(s)  = 1  + (−0.300 + 1.70i)2-s + (−1.87 − 0.684i)4-s + (0.839 − 0.704i)5-s + (−0.710 + 0.258i)7-s + (0.866 − 1.50i)8-s + (0.948 + 1.64i)10-s + (−0.565 − 0.474i)11-s + (−0.0481 − 0.273i)13-s + (−0.227 − 1.28i)14-s + (0.766 + 0.642i)16-s + (−0.891 − 1.54i)17-s + (0.114 − 0.198i)19-s + (−2.05 + 0.749i)20-s + (0.979 − 0.822i)22-s + (−0.479 − 0.174i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701746 - 0.0820224i\)
\(L(\frac12)\) \(\approx\) \(0.701746 - 0.0820224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.425 - 2.41i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-1.87 + 1.57i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.87 - 0.684i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (1.87 + 1.57i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.67 + 6.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.30 + 0.837i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.850 + 4.82i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.939 - 0.342i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.850 + 4.82i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-8.42 - 7.07i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-9.20 + 3.35i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + (-1.87 + 1.57i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.69 - 1.71i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.21 + 6.89i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-3.67 - 6.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.21 - 6.89i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.12 + 12.0i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.36 + 4.49i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.754359773377069187279286012881, −9.299814619252975785339760472285, −8.643372434430798192143376296296, −7.66817675136544411236690951514, −6.83030377137115558545114505473, −5.88949849729308489865405908914, −5.40907205408259682944893343773, −4.43260302058404045758282899383, −2.61337457262061862131272820235, −0.39185498606565115252436813155, 1.67425066860703962612759841841, 2.55284369076679863162709418970, 3.53644016501746653482806567517, 4.53976462552000741644412006268, 5.98901889903242688264675901853, 6.86393422021330070506602330635, 8.203345128394665397791655164256, 9.167433575749506248188879348841, 9.893661370661692994535946131478, 10.52560638576943268439683311850

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