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Properties

Label 2-3e6-27.22-c1-0-17
Degree $2$
Conductor $729$
Sign $0.893 + 0.448i$
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  + (−1.32 + 1.11i)2-s + (0.173 − 0.984i)4-s + (−1.62 + 0.592i)5-s + (0.347 + 1.96i)7-s + (−0.866 − 1.5i)8-s + (1.49 − 2.59i)10-s + (−3.25 − 1.18i)11-s + (−0.766 − 0.642i)13-s + (−2.65 − 2.22i)14-s + (4.69 + 1.71i)16-s + (2.59 − 4.5i)17-s + (−1 − 1.73i)19-s + (0.300 + 1.70i)20-s + (5.63 − 2.05i)22-s + (−0.601 + 3.41i)23-s + ⋯
L(s)  = 1  + (−0.938 + 0.787i)2-s + (0.0868 − 0.492i)4-s + (−0.727 + 0.264i)5-s + (0.131 + 0.744i)7-s + (−0.306 − 0.530i)8-s + (0.474 − 0.821i)10-s + (−0.981 − 0.357i)11-s + (−0.212 − 0.178i)13-s + (−0.709 − 0.595i)14-s + (1.17 + 0.427i)16-s + (0.630 − 1.09i)17-s + (−0.229 − 0.397i)19-s + (0.0672 + 0.381i)20-s + (1.20 − 0.437i)22-s + (−0.125 + 0.711i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.893 + 0.448i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402504 - 0.0953953i\)
\(L(\frac12)\) \(\approx\) \(0.402504 - 0.0953953i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.11i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.62 - 0.592i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.347 - 1.96i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (3.25 + 1.18i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.59 + 4.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.601 - 3.41i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.32 - 1.11i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.38 + 7.87i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.30 - 4.45i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.87 + 0.684i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.20 + 6.82i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-13.0 + 4.73i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.21 + 6.89i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.66 + 6.42i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (5.19 - 9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.53 + 1.28i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-10.6 + 8.90i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.87 + 0.684i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.991521626495780893517734415262, −9.350638208343330914954523855052, −8.435833160099019833627883981993, −7.69465843364561131954413711575, −7.28301308714599193714133452798, −6.01409831255299123634591723961, −5.21565396152939948687550191200, −3.71355973512997442452007726470, −2.59480105021525445966175096067, −0.33661693654214458371783424770, 1.14808522248618584994817858270, 2.50794967097364466926721323671, 3.81727831172371027334532033561, 4.83761830684622737736359246003, 6.04379869952827258845493667054, 7.43099349340776597879193163614, 8.067965565226054716729837797395, 8.698364501282562822141936219527, 9.898724052846833132139568021067, 10.41160870556686193075326022784

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