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Properties

Label 2-3e6-27.16-c1-0-26
Degree $2$
Conductor $729$
Sign $0.993 - 0.116i$
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 + 1.36i)2-s + (0.436 + 2.47i)4-s + (−1.94 − 0.708i)5-s + (0.841 − 4.77i)7-s + (−0.547 + 0.949i)8-s + (−2.20 − 3.81i)10-s + (3.89 − 1.41i)11-s + (−0.931 + 0.781i)13-s + (7.89 − 6.62i)14-s + (2.53 − 0.924i)16-s + (1.18 + 2.04i)17-s + (0.919 − 1.59i)19-s + (0.904 − 5.13i)20-s + (8.28 + 3.01i)22-s + (0.747 + 4.23i)23-s + ⋯
L(s)  = 1  + (1.15 + 0.965i)2-s + (0.218 + 1.23i)4-s + (−0.870 − 0.316i)5-s + (0.318 − 1.80i)7-s + (−0.193 + 0.335i)8-s + (−0.695 − 1.20i)10-s + (1.17 − 0.427i)11-s + (−0.258 + 0.216i)13-s + (2.10 − 1.77i)14-s + (0.634 − 0.231i)16-s + (0.286 + 0.496i)17-s + (0.210 − 0.365i)19-s + (0.202 − 1.14i)20-s + (1.76 + 0.642i)22-s + (0.155 + 0.883i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58549 + 0.150587i\)
\(L(\frac12)\) \(\approx\) \(2.58549 + 0.150587i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 1.36i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (1.94 + 0.708i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.841 + 4.77i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-3.89 + 1.41i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.931 - 0.781i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.18 - 2.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.919 + 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.747 - 4.23i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.28 + 1.91i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.255 - 1.45i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.48 + 7.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.73 - 1.45i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.15 + 1.87i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.24 - 7.07i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + (-0.246 - 0.0896i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.773 - 4.38i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.16 + 2.65i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.54 - 2.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.38 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.48 - 2.92i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.47 - 5.43i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-8.48 + 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.80 + 1.74i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.58582724534335135116353442349, −9.483689742539548557612012954786, −8.252704060244707009790265392309, −7.41734528050737598082149713253, −7.02807906462631779197671638566, −5.96290303254971651026172350231, −4.81726955374990927295747912670, −3.91726771562235788626387915410, −3.71728825421780746763513165962, −1.05093545489228402052938234353, 1.81318603691250513889698598825, 2.85560952542137306173491207601, 3.73958360455231065877674930633, 4.79269896686012763699258959412, 5.54796539812505465322230164434, 6.59916280436407715103794799136, 7.84776707313525689668015888016, 8.757196205774273326661070770668, 9.661196089260713600338528920415, 10.77335729426542941708641990446

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