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Properties

Label 2-3e6-27.13-c1-0-15
Degree $2$
Conductor $729$
Sign $0.993 + 0.116i$
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  + (−0.0300 − 0.170i)2-s + (1.85 − 0.673i)4-s + (2.86 + 2.40i)5-s + (−2.84 − 1.03i)7-s + (−0.343 − 0.594i)8-s + (0.323 − 0.559i)10-s + (1.90 − 1.60i)11-s + (−0.132 + 0.753i)13-s + (−0.0910 + 0.516i)14-s + (2.92 − 2.45i)16-s + (2.31 − 4.00i)17-s + (0.305 + 0.529i)19-s + (6.91 + 2.51i)20-s + (−0.330 − 0.277i)22-s + (6.13 − 2.23i)23-s + ⋯
L(s)  = 1  + (−0.0212 − 0.120i)2-s + (0.925 − 0.336i)4-s + (1.28 + 1.07i)5-s + (−1.07 − 0.391i)7-s + (−0.121 − 0.210i)8-s + (0.102 − 0.177i)10-s + (0.575 − 0.482i)11-s + (−0.0368 + 0.208i)13-s + (−0.0243 + 0.138i)14-s + (0.731 − 0.614i)16-s + (0.560 − 0.970i)17-s + (0.0701 + 0.121i)19-s + (1.54 + 0.562i)20-s + (−0.0703 − 0.0590i)22-s + (1.27 − 0.465i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16590 - 0.126149i\)
\(L(\frac12)\) \(\approx\) \(2.16590 - 0.126149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0300 + 0.170i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-2.86 - 2.40i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.84 + 1.03i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.90 + 1.60i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.132 - 0.753i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.31 + 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.305 - 0.529i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.13 + 2.23i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.13 - 6.45i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.15 - 2.24i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.47 - 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.913 - 5.18i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-4.26 + 3.58i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.04 - 0.378i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 8.84T + 53T^{2} \)
59 \( 1 + (-9.07 - 7.61i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.69 + 2.80i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.210 - 1.19i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.45 - 4.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.14 + 3.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.04 + 11.6i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.56 + 8.87i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-3.76 - 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.726 - 0.609i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.45717809009624387731653602251, −9.679145029096493607731354413264, −9.062149849970092467075871284785, −7.25936314893975849316245493211, −6.80347759357529878145038166203, −6.18506796506038804143441811994, −5.27994942779763905802314457720, −3.28625431428247310228502892853, −2.83833269408955384956497972314, −1.41152944153714230933293619812, 1.47757840641360789098729343560, 2.54498750295500218265423452041, 3.79838437243650465936763464009, 5.32773723860444371770967723847, 5.99472233978449131508074165683, 6.70348206235167421125576111245, 7.79172700306583375076668038803, 8.923833427282779443164282660353, 9.466442671399178664065365611299, 10.22216634222898708369920269096

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