Properties

Label 2-3e6-27.13-c1-0-0
Degree $2$
Conductor $729$
Sign $-0.835 + 0.549i$
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  + (0.300 + 1.70i)2-s + (−0.939 + 0.342i)4-s + (−1.32 − 1.11i)5-s + (−1.87 − 0.684i)7-s + (0.866 + 1.50i)8-s + (1.49 − 2.59i)10-s + (−2.65 + 2.22i)11-s + (−0.173 + 0.984i)13-s + (0.601 − 3.41i)14-s + (−3.83 + 3.21i)16-s + (−2.59 + 4.5i)17-s + (−1 − 1.73i)19-s + (1.62 + 0.592i)20-s + (−4.59 − 3.85i)22-s + (−3.25 + 1.18i)23-s + ⋯
L(s)  = 1  + (0.212 + 1.20i)2-s + (−0.469 + 0.171i)4-s + (−0.593 − 0.497i)5-s + (−0.710 − 0.258i)7-s + (0.306 + 0.530i)8-s + (0.474 − 0.821i)10-s + (−0.800 + 0.671i)11-s + (−0.0481 + 0.273i)13-s + (0.160 − 0.911i)14-s + (−0.957 + 0.803i)16-s + (−0.630 + 1.09i)17-s + (−0.229 − 0.397i)19-s + (0.363 + 0.132i)20-s + (−0.979 − 0.822i)22-s + (−0.678 + 0.247i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.835 + 0.549i$
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: \((2,\ 729,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167184 - 0.558435i\)
\(L(\frac12)\) \(\approx\) \(0.167184 - 0.558435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.300 - 1.70i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (1.32 + 1.11i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.87 + 0.684i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.65 - 2.22i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.59 - 4.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.25 - 1.18i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.300 - 1.70i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (7.51 - 2.73i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.20 - 6.82i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.53 + 1.28i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (6.51 + 2.36i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-10.6 - 8.90i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-6.57 - 2.39i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.73 - 9.84i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.19 + 9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.347 - 1.96i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.40 + 13.6i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.53 + 1.28i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.82585971676707798120650303427, −9.986011797223396158525297234519, −8.841213903909955588338328810937, −8.139015259228360777288296151511, −7.31889577915189228500937524702, −6.61613345741667317973520037462, −5.68007140144769493723470332359, −4.69693524470773182655804537694, −3.86388735675631636893864464163, −2.14280854088380972807815994625, 0.25732361612947361116452664767, 2.25357579427254747573971788132, 3.12923648066910479015380413594, 3.85767450466375343022856237533, 5.13968686910292797569788124667, 6.36126391357485455946790722139, 7.29277956039818256551040306249, 8.190677185793186205107738896923, 9.418404108676126764526282557792, 10.04293983383212639614970609897

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