Properties

Label 2-3e6-27.25-c1-0-7
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.469 + 2.66i)2-s + (−4.99 − 1.81i)4-s + (1.28 − 1.07i)5-s + (−0.470 + 0.171i)7-s + (4.48 − 7.77i)8-s + (2.26 + 3.91i)10-s + (1.46 + 1.23i)11-s + (0.540 + 3.06i)13-s + (−0.235 − 1.33i)14-s + (10.4 + 8.76i)16-s + (1.33 + 2.30i)17-s + (−2.89 + 5.02i)19-s + (−8.35 + 3.04i)20-s + (−3.97 + 3.33i)22-s + (−4.36 − 1.58i)23-s + ⋯
L(s)  = 1  + (−0.332 + 1.88i)2-s + (−2.49 − 0.909i)4-s + (0.572 − 0.480i)5-s + (−0.177 + 0.0647i)7-s + (1.58 − 2.74i)8-s + (0.715 + 1.23i)10-s + (0.442 + 0.371i)11-s + (0.149 + 0.850i)13-s + (−0.0628 − 0.356i)14-s + (2.61 + 2.19i)16-s + (0.323 + 0.559i)17-s + (−0.664 + 1.15i)19-s + (−1.86 + 0.680i)20-s + (−0.847 + 0.710i)22-s + (−0.910 − 0.331i)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: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0579920 - 0.995685i\)
\(L(\frac12)\) \(\approx\) \(0.0579920 - 0.995685i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.469 - 2.66i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-1.28 + 1.07i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.470 - 0.171i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.46 - 1.23i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.540 - 3.06i)T + (-12.2 + 4.44i)T^{2} \)
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 \( 1 + (4.36 + 1.58i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.454 + 2.57i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.33 - 1.57i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.42 - 4.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.00 - 11.3i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.89 + 5.78i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.42 + 2.33i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (-1.67 + 1.40i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (6.42 - 2.33i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.16 - 12.2i)T + (-62.9 + 22.9i)T^{2} \)
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 \( 1 + (-0.922 + 5.23i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.473 + 2.68i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-5.60 + 9.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.27 + 4.42i)T + (16.8 + 95.5i)T^{2} \)
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

−10.12209962226809847652930026486, −9.808234014888545804975775151939, −8.793290174744100515833447753102, −8.284118921044668454864190233464, −7.30813848271691652174228969350, −6.26205090785420892340938799400, −5.97411616839765404687307896977, −4.75878126727127969688135375537, −3.98972056681168584135185840453, −1.50905886821504996733171520773, 0.62559665007209095477646932732, 2.14399182281148285816232539245, 2.98977374024965697366891521406, 3.94945897183421493148540456643, 5.09024773989120760982744834484, 6.27852030982262400916334892745, 7.69226485421097471495311524247, 8.681268566097626868384413996344, 9.392302041880616363721857136497, 10.18559781700830130775235470391

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