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Properties

Label 2-3e6-27.22-c1-0-12
Degree $2$
Conductor $729$
Sign $0.973 - 0.230i$
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  + (1.32 − 1.11i)2-s + (0.173 − 0.984i)4-s + (−3.25 + 1.18i)5-s + (−0.173 − 0.984i)7-s + (0.866 + 1.5i)8-s + (−2.99 + 5.19i)10-s + (3.25 + 1.18i)11-s + (3.83 + 3.21i)13-s + (−1.32 − 1.11i)14-s + (4.69 + 1.71i)16-s + (0.5 + 0.866i)19-s + (0.601 + 3.41i)20-s + (5.63 − 2.05i)22-s + (−1.20 + 6.82i)23-s + (5.36 − 4.49i)25-s + 8.66·26-s + ⋯
L(s)  = 1  + (0.938 − 0.787i)2-s + (0.0868 − 0.492i)4-s + (−1.45 + 0.529i)5-s + (−0.0656 − 0.372i)7-s + (0.306 + 0.530i)8-s + (−0.948 + 1.64i)10-s + (0.981 + 0.357i)11-s + (1.06 + 0.891i)13-s + (−0.354 − 0.297i)14-s + (1.17 + 0.427i)16-s + (0.114 + 0.198i)19-s + (0.134 + 0.762i)20-s + (1.20 − 0.437i)22-s + (−0.250 + 1.42i)23-s + (1.07 − 0.899i)25-s + 1.69·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.973 - 0.230i$
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: \((2,\ 729,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00152 + 0.233944i\)
\(L(\frac12)\) \(\approx\) \(2.00152 + 0.233944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 1.11i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (3.25 - 1.18i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.173 + 0.984i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-3.25 - 1.18i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.83 - 3.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.20 - 6.82i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.65 - 2.22i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.868 + 4.92i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.65 - 2.22i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.601 + 3.41i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (3.25 - 1.18i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.347 - 1.96i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.12 - 5.14i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (5.19 - 9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-5.30 + 4.45i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.9 + 5.81i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.99240192199723025247786256962, −9.771963737582650577831880997306, −8.661740185661854720902801111476, −7.70261231713804902003681264732, −6.99371076027031142677592141653, −5.83526346379095360886278471794, −4.34340763773374538271895667992, −3.93504296160723579282892074090, −3.22466258313304849425763074670, −1.60089700235212566798291976901, 0.879358233989706670386148057197, 3.30676864628407184998818128959, 4.05091152630573428446716415788, 4.84908407161049555593192179699, 5.93587246432539520278571581533, 6.63858884065404074318227342611, 7.73032139181822656915942430002, 8.380373163571824980782950017816, 9.220904426934283104555323776336, 10.58556212899554608720512965841

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