Properties

Label 2-3e6-27.22-c1-0-20
Degree $2$
Conductor $729$
Sign $0.802 + 0.597i$
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  + (0.984 − 0.826i)2-s + (−0.0603 + 0.342i)4-s + (−0.419 + 0.152i)5-s + (−0.613 − 3.47i)7-s + (1.50 + 2.61i)8-s + (−0.286 + 0.497i)10-s + (2.61 + 0.950i)11-s + (2.52 + 2.11i)13-s + (−3.47 − 2.91i)14-s + (2.99 + 1.08i)16-s + (3.51 − 6.09i)17-s + (2.59 + 4.49i)19-s + (−0.0269 − 0.152i)20-s + (3.35 − 1.22i)22-s + (1.26 − 7.16i)23-s + ⋯
L(s)  = 1  + (0.696 − 0.584i)2-s + (−0.0301 + 0.171i)4-s + (−0.187 + 0.0682i)5-s + (−0.231 − 1.31i)7-s + (0.533 + 0.923i)8-s + (−0.0907 + 0.157i)10-s + (0.787 + 0.286i)11-s + (0.699 + 0.586i)13-s + (−0.929 − 0.780i)14-s + (0.748 + 0.272i)16-s + (0.853 − 1.47i)17-s + (0.594 + 1.03i)19-s + (−0.00602 − 0.0341i)20-s + (0.716 − 0.260i)22-s + (0.263 − 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.802 + 0.597i$
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: \((2,\ 729,\ (\ :1/2),\ 0.802 + 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14988 - 0.712392i\)
\(L(\frac12)\) \(\approx\) \(2.14988 - 0.712392i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.984 + 0.826i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.419 - 0.152i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.613 + 3.47i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.61 - 0.950i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-2.52 - 2.11i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-3.51 + 6.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.26 + 7.16i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.77 + 2.32i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.336 - 1.90i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.72 + 3.12i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-5.41 - 1.96i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.524 - 2.97i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 8.77T + 53T^{2} \)
59 \( 1 + (2.78 - 1.01i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.36 - 7.76i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.23 + 6.06i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.65 - 4.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.777 - 1.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.11 - 7.65i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (12.4 - 10.4i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (9.21 + 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.50 + 3.45i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.46005068344180797121284721529, −9.663685616604144246519390747293, −8.575742106743886083951904829904, −7.52132144461606729682367730769, −6.96603775748770922182812347721, −5.65696512349237488563373005895, −4.36201277429837136479235391877, −3.91183845130287055121395234331, −2.88943876020430479712897705717, −1.24849247828280817944190120502, 1.37020262709772432375377877964, 3.16115876209825495011770544431, 4.09328610360262878746658805083, 5.44251723604598403512822382705, 5.82107074045175769759228703364, 6.66048248269546008898950410490, 7.82503139452889782571474149045, 8.749160502438549675213820409435, 9.535789808248683396361776633709, 10.42753289283244628493288452398

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