↑

Properties

Label 2-3e6-27.13-c1-0-16
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: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0579920 + 0.995685i\)
\(L(\frac12)\) \(\approx\) \(0.0579920 + 0.995685i\)
\(L(\frac{3}{2})\) 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

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

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