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Properties

Label 2-3e6-27.7-c1-0-11
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  + (−1.62 + 0.592i)2-s + (0.766 − 0.642i)4-s + (−0.300 − 1.70i)5-s + (1.53 + 1.28i)7-s + (0.866 − 1.50i)8-s + (1.5 + 2.59i)10-s + (−0.601 + 3.41i)11-s + (0.939 + 0.342i)13-s + (−3.25 − 1.18i)14-s + (−0.868 + 4.92i)16-s + (−2.59 − 4.5i)17-s + (−1 + 1.73i)19-s + (−1.32 − 1.11i)20-s + (−1.04 − 5.90i)22-s + (2.65 − 2.22i)23-s + ⋯
L(s)  = 1  + (−1.15 + 0.418i)2-s + (0.383 − 0.321i)4-s + (−0.134 − 0.762i)5-s + (0.579 + 0.485i)7-s + (0.306 − 0.530i)8-s + (0.474 + 0.821i)10-s + (−0.181 + 1.02i)11-s + (0.260 + 0.0948i)13-s + (−0.869 − 0.316i)14-s + (−0.217 + 1.23i)16-s + (−0.630 − 1.09i)17-s + (−0.229 + 0.397i)19-s + (−0.296 − 0.248i)20-s + (−0.222 − 1.25i)22-s + (0.553 − 0.464i)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: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.893 - 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.785449 + 0.186154i\)
\(L(\frac12)\) \(\approx\) \(0.785449 + 0.186154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.592i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.300 + 1.70i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.53 - 1.28i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.601 - 3.41i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T + (9.95 + 8.35i)T^{2} \)
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 \( 1 + (-2.65 + 2.22i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.62 - 0.592i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-6.12 + 5.14i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.51 - 2.36i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.347 + 1.96i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.30 - 4.45i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-2.40 - 13.6i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.36 + 4.49i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-9.39 - 3.42i)T + (51.3 + 43.0i)T^{2} \)
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 \( 1 + (1.87 - 0.684i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-13.0 + 4.73i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.59 + 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.347 + 1.96i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.14232574051126502634620433117, −9.354807691930791481467375233976, −8.760797897131525563042413980916, −8.029674400269208006448300644950, −7.27745445293694764932596184306, −6.29988835795092644980494290286, −4.92449260219374964288383748813, −4.30497949101823114415310796759, −2.37577849501833945058197406608, −0.942602424970745451216896795516, 0.900752997348352219079508655158, 2.30922252866518167977126876976, 3.54481088504721843867996240706, 4.83393439557923513683416203369, 6.06446580540033435691991764915, 7.11671535361540543328282503485, 8.006004257064976461115722214286, 8.623606804774169742589535000896, 9.459719122780327452256448405724, 10.63994363549316905181273609462

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