Properties

Label 2-3e6-9.4-c1-0-29
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: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242788 + 0.420522i\)
\(L(\frac12)\) \(\approx\) \(0.242788 + 0.420522i\)
\(L(\frac{3}{2})\) 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

−9.706935382931281887884083585774, −9.361852270050604556357207654067, −8.440363170859426261349113852742, −7.24621705723239879136822270498, −6.29974857291433732504730793388, −5.23084125637617686798332767427, −4.02433631654778677108146228328, −2.85493689510042416067771745417, −1.64027894837032147625556708545, −0.28304867761566452197403852949, 2.47664318401442546730100946568, 3.15854857074466871329998084594, 5.18469828849508040270382034299, 5.70717620397806226239799613905, 6.85563724720151487365306615596, 7.37710024240139748431821414990, 8.234668896804261396438918476492, 9.222501686218024692427092091158, 9.918783730101965551322262920689, 10.58974025686747428158145442267

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