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Properties

Label 2-3e6-9.7-c1-0-22
Degree $2$
Conductor $729$
Sign $-0.500 + 0.866i$
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  + (0.789 − 1.36i)2-s + (−0.245 − 0.425i)4-s + (−0.839 − 1.45i)5-s + (−1.38 + 2.40i)7-s + 2.38·8-s − 2.64·10-s + (2.07 − 3.59i)11-s + (−3.43 − 5.95i)13-s + (2.19 + 3.79i)14-s + (2.37 − 4.10i)16-s + 0.976·17-s + 2.68·19-s + (−0.412 + 0.714i)20-s + (−3.27 − 5.67i)22-s + (−0.806 − 1.39i)23-s + ⋯
L(s)  = 1  + (0.558 − 0.966i)2-s + (−0.122 − 0.212i)4-s + (−0.375 − 0.650i)5-s + (−0.525 + 0.909i)7-s + 0.841·8-s − 0.837·10-s + (0.625 − 1.08i)11-s + (−0.953 − 1.65i)13-s + (0.586 + 1.01i)14-s + (0.592 − 1.02i)16-s + 0.236·17-s + 0.616·19-s + (−0.0922 + 0.159i)20-s + (−0.698 − 1.20i)22-s + (−0.168 − 0.291i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946102 - 1.63869i\)
\(L(\frac12)\) \(\approx\) \(0.946102 - 1.63869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.789 + 1.36i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.839 + 1.45i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.38 - 2.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.07 + 3.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.43 + 5.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.976T + 17T^{2} \)
19 \( 1 - 2.68T + 19T^{2} \)
23 \( 1 + (0.806 + 1.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.11 + 7.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.522 + 0.904i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.30T + 37T^{2} \)
41 \( 1 + (-2.42 - 4.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.92 - 8.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.24 - 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 + (4.52 + 7.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.642 + 1.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.62T + 71T^{2} \)
73 \( 1 + 4.56T + 73T^{2} \)
79 \( 1 + (2.32 - 4.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.88 - 4.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.54T + 89T^{2} \)
97 \( 1 + (4.28 - 7.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.16973411163349176283440503670, −9.497587550603870875162348884859, −8.296865467827726451850856202944, −7.82553938535047285889237964131, −6.31519734567209426612045757689, −5.38198892248509015980590807173, −4.46652090688884420240186418068, −3.23498525246339654134147386759, −2.66641481971031820711569172482, −0.861854829186206914209836506245, 1.77773585070207320184904259370, 3.55663779443170284953534351810, 4.38572239102599379907840421201, 5.26747138655660053303948543433, 6.76693400797861510924447661196, 6.95403443905955133026909207129, 7.44171320839708280566897753509, 8.940642457379744883416576766008, 9.943943423297730858292824284031, 10.49322540189118764074849466803

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