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Properties

Label 2-3e6-27.7-c1-0-9
Degree $2$
Conductor $729$
Sign $0.396 - 0.918i$
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  + (1.48 − 0.539i)2-s + (0.376 − 0.315i)4-s + (0.291 + 1.65i)5-s + (2.12 + 1.78i)7-s + (−1.19 + 2.06i)8-s + (1.32 + 2.29i)10-s + (−0.720 + 4.08i)11-s + (−6.46 − 2.35i)13-s + (4.12 + 1.50i)14-s + (−0.823 + 4.66i)16-s + (−0.488 − 0.845i)17-s + (−1.34 + 2.32i)19-s + (0.631 + 0.530i)20-s + (1.13 + 6.45i)22-s + (1.23 − 1.03i)23-s + ⋯
L(s)  = 1  + (1.04 − 0.381i)2-s + (0.188 − 0.157i)4-s + (0.130 + 0.739i)5-s + (0.804 + 0.675i)7-s + (−0.420 + 0.729i)8-s + (0.418 + 0.725i)10-s + (−0.217 + 1.23i)11-s + (−1.79 − 0.652i)13-s + (1.10 + 0.400i)14-s + (−0.205 + 1.16i)16-s + (−0.118 − 0.205i)17-s + (−0.308 + 0.533i)19-s + (0.141 + 0.118i)20-s + (0.242 + 1.37i)22-s + (0.257 − 0.216i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.396 - 0.918i$
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: \((2,\ 729,\ (\ :1/2),\ 0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91912 + 1.26223i\)
\(L(\frac12)\) \(\approx\) \(1.91912 + 1.26223i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 0.539i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.291 - 1.65i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.12 - 1.78i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.720 - 4.08i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (6.46 + 2.35i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.488 + 0.845i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.34 - 2.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.23 + 1.03i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.73 + 2.81i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.799 + 0.671i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-0.654 - 1.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.55 - 1.65i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.70 + 9.69i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-9.57 - 8.03i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 + (-1.57 - 8.91i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.984 + 0.826i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-4.36 - 1.58i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.81 + 4.87i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.28 + 3.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.37 - 1.59i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (5.41 - 1.97i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.27 - 3.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.48 + 8.44i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.60485154305179946818919131612, −9.986159882025040259355261187630, −8.830212496731382885046127377896, −7.83725515767528760850381347826, −7.02456558383984572914260044187, −5.77401038957737658641690973340, −4.93216155978932457258973896641, −4.32079662736700970248130047567, −2.67231822097827798918205491854, −2.36285054032339277545674111734, 0.862406178832802408699942016896, 2.72692434861609898035519610321, 4.14874518961923245377045604676, 4.82293705255855980202062502705, 5.41885683204512809527986700231, 6.60235106824177818794161718301, 7.41257573946087399874192312425, 8.511812529250386752607792882796, 9.282303777760796020361530094944, 10.28294949316347301006994716862

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