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Properties

Label 2-3e6-27.16-c1-0-5
Degree $2$
Conductor $729$
Sign $0.597 + 0.802i$
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  + (−2.07 − 1.73i)2-s + (0.923 + 5.23i)4-s + (−1.57 − 0.571i)5-s + (0.0869 − 0.492i)7-s + (4.48 − 7.77i)8-s + (2.26 + 3.91i)10-s + (−1.80 + 0.655i)11-s + (2.38 − 2.00i)13-s + (−1.03 + 0.870i)14-s + (−12.8 + 4.66i)16-s + (1.33 + 2.30i)17-s + (−2.89 + 5.02i)19-s + (1.54 − 8.75i)20-s + (4.87 + 1.77i)22-s + (0.806 + 4.57i)23-s + ⋯
L(s)  = 1  + (−1.46 − 1.22i)2-s + (0.461 + 2.61i)4-s + (−0.702 − 0.255i)5-s + (0.0328 − 0.186i)7-s + (1.58 − 2.74i)8-s + (0.715 + 1.23i)10-s + (−0.543 + 0.197i)11-s + (0.661 − 0.554i)13-s + (−0.277 + 0.232i)14-s + (−3.20 + 1.16i)16-s + (0.323 + 0.559i)17-s + (−0.664 + 1.15i)19-s + (0.345 − 1.95i)20-s + (1.03 + 0.378i)22-s + (0.168 + 0.954i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.597 + 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500910 - 0.251566i\)
\(L(\frac12)\) \(\approx\) \(0.500910 - 0.251566i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 1.73i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (1.57 + 0.571i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.0869 + 0.492i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.80 - 0.655i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-2.38 + 2.00i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.33 - 2.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.89 - 5.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.806 - 4.57i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.00 - 1.68i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.801 + 4.54i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-2.42 - 4.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.84 + 7.42i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-8.46 + 3.07i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.18 - 6.72i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (2.05 + 0.749i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.18 + 6.73i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.56 + 8.02i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.41 + 2.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.96 - 8.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.06 - 3.41i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-2.08 - 1.75i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-5.60 + 9.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.47 + 2.35i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.37969683884155445584940889861, −9.524830828450764960757048847507, −8.602812345346079378413657812036, −7.905553028944050932594733875872, −7.48854660202434090416809255269, −5.94839508516316418626097329951, −4.17485654293844077104328216957, −3.49135180146679559501383242786, −2.19075488107521495113827462870, −0.856026064690889539807907368939, 0.70496088058976023880483058508, 2.48883188895818904739972440295, 4.39156666486890988622286020428, 5.53274588292742273090108036341, 6.46276502525247436928956716846, 7.20629906278798102201978192227, 7.930689185713944187076895738573, 8.748411718300445132308579788705, 9.279703891127326219590388940441, 10.40067109418360882111314006143

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