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Properties

Label 2-3e6-27.22-c1-0-2
Degree $2$
Conductor $729$
Sign $-0.286 + 0.957i$
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  + (−1.87 + 1.57i)2-s + (0.694 − 3.93i)4-s + (−2.30 + 0.837i)5-s + (0.347 + 1.96i)7-s + (2.44 + 4.24i)8-s + (2.99 − 5.19i)10-s + (2.30 + 0.837i)11-s + (−0.766 − 0.642i)13-s + (−3.75 − 3.14i)14-s + (−3.75 − 1.36i)16-s + (−3.67 + 6.36i)17-s + (0.5 + 0.866i)19-s + (1.70 + 9.64i)20-s + (−5.63 + 2.05i)22-s + (0.425 − 2.41i)23-s + ⋯
L(s)  = 1  + (−1.32 + 1.11i)2-s + (0.347 − 1.96i)4-s + (−1.02 + 0.374i)5-s + (0.131 + 0.744i)7-s + (0.866 + 1.50i)8-s + (0.948 − 1.64i)10-s + (0.694 + 0.252i)11-s + (−0.212 − 0.178i)13-s + (−1.00 − 0.841i)14-s + (−0.939 − 0.342i)16-s + (−0.891 + 1.54i)17-s + (0.114 + 0.198i)19-s + (0.380 + 2.15i)20-s + (−1.20 + 0.437i)22-s + (0.0886 − 0.502i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.286 + 0.957i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.286 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0910404 - 0.122288i\)
\(L(\frac12)\) \(\approx\) \(0.0910404 - 0.122288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.87 - 1.57i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (2.30 - 0.837i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.347 - 1.96i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.30 - 0.837i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (3.67 - 6.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.425 + 2.41i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.75 + 3.14i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.173 - 0.984i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.75 + 3.14i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (10.3 + 3.76i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.70 + 9.64i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + (2.30 - 0.837i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.868 - 4.92i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.36 + 4.49i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-3.67 + 6.36i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.36 - 4.49i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-9.38 + 7.87i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.57 - 2.39i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.62256439838118509478025720920, −10.03393450575412174557456312777, −8.824301623934709159704433702682, −8.491111436658245951399547279377, −7.68499287276254869216157832052, −6.73537061074135413371563153501, −6.20767850818970927245116129578, −4.93058667722799547953657044292, −3.61381099416716000138896045441, −1.80280562456634389175073435232, 0.13300248356930204438894034282, 1.32740546626422994784308053203, 2.87677162167449744720379689589, 3.88348160621524081788022795972, 4.82872292518537217066579284612, 6.79637048721170274159224038277, 7.49906912769170327326363715229, 8.292738554791100787037972816937, 9.093162180899953062096062934541, 9.698254910424481587176310110212

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