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Properties

Label 2-3e6-27.7-c1-0-8
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  + (−2.30 + 0.840i)2-s + (3.09 − 2.59i)4-s + (0.534 + 3.03i)5-s + (−2.03 − 1.70i)7-s + (−2.50 + 4.34i)8-s + (−3.78 − 6.55i)10-s + (0.596 − 3.38i)11-s + (3.14 + 1.14i)13-s + (6.14 + 2.23i)14-s + (0.737 − 4.18i)16-s + (1.28 + 2.22i)17-s + (1.04 − 1.81i)19-s + (9.53 + 8.00i)20-s + (1.46 + 8.31i)22-s + (−0.409 + 0.343i)23-s + ⋯
L(s)  = 1  + (−1.63 + 0.594i)2-s + (1.54 − 1.29i)4-s + (0.239 + 1.35i)5-s + (−0.769 − 0.645i)7-s + (−0.886 + 1.53i)8-s + (−1.19 − 2.07i)10-s + (0.179 − 1.01i)11-s + (0.871 + 0.317i)13-s + (1.64 + 0.597i)14-s + (0.184 − 1.04i)16-s + (0.311 + 0.540i)17-s + (0.240 − 0.416i)19-s + (2.13 + 1.78i)20-s + (0.312 + 1.77i)22-s + (−0.0853 + 0.0716i)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

\(L(1)\) \(\approx\) \(0.566959 + 0.372895i\)
\(L(\frac12)\) \(\approx\) \(0.566959 + 0.372895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.30 - 0.840i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.534 - 3.03i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.03 + 1.70i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.596 + 3.38i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.14 - 1.14i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.28 - 2.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.04 + 1.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.409 - 0.343i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.37 + 0.865i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.90 + 4.95i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-5.14 - 8.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.59 + 1.67i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.476 + 2.69i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.33 - 3.63i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (0.287 + 1.62i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-11.0 - 9.23i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.52 - 2.01i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.40 - 12.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.940 - 1.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (16.1 - 5.88i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3.72 + 1.35i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.84 - 10.4i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.12973548314328258670051277040, −9.931269254663846834468318509872, −8.748828955314138780025180078031, −8.053680467083032780442179696323, −7.03635592355188268255763257066, −6.46639994753677844257397556255, −5.94696812139081971593531656571, −3.77840548581334366393767928802, −2.68775533067505210576870344900, −0.978698410190728036486002141232, 0.827941321706180592468378860995, 1.95172902905279583450110084672, 3.22846924641458681058021220027, 4.76041583772071595565836404732, 5.94069746172615853232536741141, 7.04370181594678233216808848010, 8.129546232797252600093077810215, 8.690517327928718673868271329806, 9.518049377496882699348167625961, 9.791664719901709372373678373842

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