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Properties

Label 2-3e6-9.4-c1-0-14
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  + (−1.05 − 1.83i)2-s + (−1.24 + 2.14i)4-s + (−1.34 + 2.32i)5-s + (−0.486 − 0.842i)7-s + 1.01·8-s + 5.67·10-s + (−0.158 − 0.274i)11-s + (0.757 − 1.31i)13-s + (−1.02 + 1.78i)14-s + (1.40 + 2.43i)16-s − 1.17·17-s + 6.22·19-s + (−3.32 − 5.76i)20-s + (−0.335 + 0.580i)22-s + (1.08 − 1.87i)23-s + ⋯
L(s)  = 1  + (−0.748 − 1.29i)2-s + (−0.620 + 1.07i)4-s + (−0.599 + 1.03i)5-s + (−0.183 − 0.318i)7-s + 0.359·8-s + 1.79·10-s + (−0.0477 − 0.0827i)11-s + (0.209 − 0.363i)13-s + (−0.275 + 0.476i)14-s + (0.351 + 0.608i)16-s − 0.284·17-s + 1.42·19-s + (−0.743 − 1.28i)20-s + (−0.0715 + 0.123i)22-s + (0.225 − 0.390i)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: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370233 - 0.641263i\)
\(L(\frac12)\) \(\approx\) \(0.370233 - 0.641263i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 + 1.83i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.34 - 2.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.486 + 0.842i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.158 + 0.274i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.757 + 1.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 - 6.22T + 19T^{2} \)
23 \( 1 + (-1.08 + 1.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.20 + 3.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.33 + 7.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.46T + 37T^{2} \)
41 \( 1 + (-2.92 + 5.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.79 + 4.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.23 + 2.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (0.862 - 1.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.507 + 0.878i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.428 - 0.741i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.59T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + (-5.60 - 9.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.34 - 4.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + (-2.77 - 4.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.26362393606000322266413296303, −9.569668327277850238158680513230, −8.575928517986612756512254199570, −7.66897493515549359922717554809, −6.86393047771686290129406160942, −5.64213702165844004233337554964, −3.98389269901978673705985206526, −3.26057461790872879897873707219, −2.31218274410756333973911624608, −0.61391397650232916310165313531, 1.09588014930606698960080929459, 3.23993956856859068121671646211, 4.71136522508940279676033546619, 5.42241843063361283140282662324, 6.47196631654845252732713230648, 7.35927634411976562476862134363, 8.071048187867451435502491601135, 8.926671395904681518159716824223, 9.263140620024646259237145315507, 10.34825548123885169750956915492

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