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Properties

Label 2-3e6-729.4-c1-0-33
Degree $2$
Conductor $729$
Sign $0.900 + 0.434i$
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  + (−2.03 − 0.0263i)2-s + (−1.63 + 0.562i)3-s + (2.15 + 0.0556i)4-s + (0.558 + 0.171i)5-s + (3.35 − 1.10i)6-s + (0.399 − 0.997i)7-s + (−0.311 − 0.0120i)8-s + (2.36 − 1.84i)9-s + (−1.13 − 0.363i)10-s + (4.38 + 2.49i)11-s + (−3.55 + 1.11i)12-s + (−0.846 − 1.89i)13-s + (−0.841 + 2.02i)14-s + (−1.01 + 0.0339i)15-s + (−3.66 − 0.189i)16-s + (−0.958 − 0.435i)17-s + ⋯
L(s)  = 1  + (−1.44 − 0.0186i)2-s + (−0.945 + 0.324i)3-s + (1.07 + 0.0278i)4-s + (0.249 + 0.0765i)5-s + (1.36 − 0.450i)6-s + (0.151 − 0.376i)7-s + (−0.109 − 0.00426i)8-s + (0.788 − 0.614i)9-s + (−0.358 − 0.114i)10-s + (1.32 + 0.752i)11-s + (−1.02 + 0.323i)12-s + (−0.234 − 0.525i)13-s + (−0.224 + 0.540i)14-s + (−0.261 + 0.00875i)15-s + (−0.916 − 0.0474i)16-s + (−0.232 − 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531594 - 0.121542i\)
\(L(\frac12)\) \(\approx\) \(0.531594 - 0.121542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 - 0.562i)T \)
good2 \( 1 + (2.03 + 0.0263i)T + (1.99 + 0.0517i)T^{2} \)
5 \( 1 + (-0.558 - 0.171i)T + (4.14 + 2.80i)T^{2} \)
7 \( 1 + (-0.399 + 0.997i)T + (-5.06 - 4.83i)T^{2} \)
11 \( 1 + (-4.38 - 2.49i)T + (5.62 + 9.45i)T^{2} \)
13 \( 1 + (0.846 + 1.89i)T + (-8.67 + 9.68i)T^{2} \)
17 \( 1 + (0.958 + 0.435i)T + (11.1 + 12.8i)T^{2} \)
19 \( 1 + (0.947 + 1.38i)T + (-6.84 + 17.7i)T^{2} \)
23 \( 1 + (0.543 - 1.65i)T + (-18.5 - 13.6i)T^{2} \)
29 \( 1 + (-5.87 + 3.75i)T + (12.1 - 26.3i)T^{2} \)
31 \( 1 + (0.116 + 0.428i)T + (-26.7 + 15.6i)T^{2} \)
37 \( 1 + (4.10 + 0.806i)T + (34.2 + 13.9i)T^{2} \)
41 \( 1 + (3.84 - 1.45i)T + (30.7 - 27.1i)T^{2} \)
43 \( 1 + (-5.40 + 2.37i)T + (29.1 - 31.6i)T^{2} \)
47 \( 1 + (-1.43 + 5.29i)T + (-40.5 - 23.7i)T^{2} \)
53 \( 1 + (5.29 - 5.61i)T + (-3.08 - 52.9i)T^{2} \)
59 \( 1 + (-3.64 + 0.619i)T + (55.6 - 19.4i)T^{2} \)
61 \( 1 + (-4.15 + 7.64i)T + (-33.1 - 51.1i)T^{2} \)
67 \( 1 + (-11.8 + 6.16i)T + (38.6 - 54.7i)T^{2} \)
71 \( 1 + (-11.2 - 8.68i)T + (17.7 + 68.7i)T^{2} \)
73 \( 1 + (3.21 - 1.03i)T + (59.3 - 42.4i)T^{2} \)
79 \( 1 + (-2.66 + 14.0i)T + (-73.5 - 28.9i)T^{2} \)
83 \( 1 + (2.50 + 15.3i)T + (-78.7 + 26.3i)T^{2} \)
89 \( 1 + (-16.0 - 6.54i)T + (63.5 + 62.3i)T^{2} \)
97 \( 1 + (-0.177 + 0.772i)T + (-87.2 - 42.4i)T^{2} \)
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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

−10.14514422582597520406731684047, −9.658896689355263271462383559861, −8.870540271218907673406543016126, −7.75731948847400656848072181986, −6.92053152084429239855286796211, −6.24596518111184430831500615206, −4.88383150392220459133903076124, −3.98037794866503467797889458049, −1.96180510432492821899928345939, −0.68556052348587136150644904321, 1.01718718266585934705006215863, 2.03922797758491225964716672175, 4.06014884236662301604911638022, 5.29344552963585276321153012691, 6.42420089307109301291158163680, 6.89342084027173291534803068984, 8.055401683310247083201388448116, 8.808271661896056075655494631634, 9.551507322256283713785509294901, 10.37264715815302181431141037530

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