↑

Properties

Label 2-3e6-9.7-c1-0-0
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

Related objects

Downloads

Learn more

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

\(L(1)\) \(\approx\) \(0.242788 - 0.420522i\)
\(L(\frac12)\) \(\approx\) \(0.242788 - 0.420522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.28 - 7.42i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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