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Properties

Label 2-3e6-27.7-c1-0-26
Degree $2$
Conductor $729$
Sign $0.597 + 0.802i$
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  + (1.99 − 0.726i)2-s + (1.92 − 1.61i)4-s + (−0.359 − 2.03i)5-s + (3.71 + 3.11i)7-s + (0.547 − 0.949i)8-s + (−2.20 − 3.81i)10-s + (0.720 − 4.08i)11-s + (1.14 + 0.415i)13-s + (9.68 + 3.52i)14-s + (−0.469 + 2.66i)16-s + (−1.18 − 2.04i)17-s + (0.919 − 1.59i)19-s + (−3.99 − 3.34i)20-s + (−1.53 − 8.68i)22-s + (−3.29 + 2.76i)23-s + ⋯
L(s)  = 1  + (1.41 − 0.513i)2-s + (0.963 − 0.808i)4-s + (−0.160 − 0.912i)5-s + (1.40 + 1.17i)7-s + (0.193 − 0.335i)8-s + (−0.695 − 1.20i)10-s + (0.217 − 1.23i)11-s + (0.316 + 0.115i)13-s + (2.58 + 0.941i)14-s + (−0.117 + 0.665i)16-s + (−0.286 − 0.496i)17-s + (0.210 − 0.365i)19-s + (−0.892 − 0.748i)20-s + (−0.326 − 1.85i)22-s + (−0.687 + 0.576i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.597 + 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.13868 - 1.57630i\)
\(L(\frac12)\) \(\approx\) \(3.13868 - 1.57630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 + 0.726i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.359 + 2.03i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-3.71 - 3.11i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.720 + 4.08i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-1.14 - 0.415i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.18 + 2.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.919 + 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.29 - 2.76i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (2.80 - 1.01i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.12 + 0.947i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (4.48 + 7.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.12 + 0.773i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.952 - 5.40i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.50 - 4.61i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + (-0.0455 - 0.258i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (3.41 + 2.86i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.88 + 1.41i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (1.54 + 2.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.38 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.27 - 1.55i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.94 + 2.89i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (8.48 - 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.887 - 5.03i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.89089635642108474162902505128, −9.129781361553104965330947278269, −8.678922500157223953605774763936, −7.81743924003950401067483465070, −6.16186965778636064913654334708, −5.42577146466280926517624967951, −4.87083457646151190641160781975, −3.90237060006861297329408920039, −2.67423941281932973239710249762, −1.50562230357598342551860893109, 1.82881475737310755686955108373, 3.38756370988633368751773359421, 4.28774885263620436888773361740, 4.83585441909194593273983548962, 6.05805260858870319692215412441, 7.05302324574719254755246654953, 7.39936768491866321137372128691, 8.444041649033216710175235404231, 10.04928360137827523066991451178, 10.61824638687717921313435547341

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