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Properties

Label 2-3e6-9.7-c1-0-26
Degree $2$
Conductor $729$
Sign $0.5 + 0.866i$
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  + (−0.642 + 1.11i)2-s + (0.173 + 0.300i)4-s + (−0.223 − 0.386i)5-s + (1.76 − 3.05i)7-s − 3.01·8-s + 0.573·10-s + (1.39 − 2.40i)11-s + (−1.64 − 2.84i)13-s + (2.27 + 3.93i)14-s + (1.59 − 2.75i)16-s − 7.03·17-s − 5.18·19-s + (0.0775 − 0.134i)20-s + (1.78 + 3.09i)22-s + (−3.63 − 6.30i)23-s + ⋯
L(s)  = 1  + (−0.454 + 0.787i)2-s + (0.0868 + 0.150i)4-s + (−0.0998 − 0.172i)5-s + (0.667 − 1.15i)7-s − 1.06·8-s + 0.181·10-s + (0.419 − 0.725i)11-s + (−0.456 − 0.790i)13-s + (0.606 + 1.05i)14-s + (0.398 − 0.689i)16-s − 1.70·17-s − 1.18·19-s + (0.0173 − 0.0300i)20-s + (0.380 + 0.659i)22-s + (−0.758 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.5 + 0.866i$
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: \((2,\ 729,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673403 - 0.388789i\)
\(L(\frac12)\) \(\approx\) \(0.673403 - 0.388789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.642 - 1.11i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.223 + 0.386i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.76 + 3.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.39 + 2.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.64 + 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.03T + 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 + (3.63 + 6.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.80 - 3.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.967 - 1.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.22T + 37T^{2} \)
41 \( 1 + (-2.43 - 4.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.87 + 4.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.50 - 2.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.77T + 53T^{2} \)
59 \( 1 + (1.48 + 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.94 - 6.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.71 - 8.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.30T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + (-5.95 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.12 + 14.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 + (5.05 - 8.75i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.43399789135746425900272692576, −8.928623376723712927319342800017, −8.485173787475932399604272171384, −7.70665552347173048938236252814, −6.79375937798040360181947359536, −6.19760102259858875815566083982, −4.75360162840912113795854295719, −3.90206673212008304593016614752, −2.43773321636966622664414556401, −0.43283030403012660915918717608, 1.92977482336545821204068274227, 2.27576372019749432269423232845, 3.96463361327423769417072945992, 5.07861016009489820502712212287, 6.16507050406569318014053872517, 6.97883824569037124749857321680, 8.259629991520783241445505505905, 9.197571352045883377815194287755, 9.465325140351665468835459439485, 10.71995943106976382510554044661

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