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Properties

Label 2-3e6-27.22-c1-0-32
Degree $2$
Conductor $729$
Sign $-0.993 - 0.116i$
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  + (1.88 − 1.57i)2-s + (0.701 − 3.97i)4-s + (−2.89 + 1.05i)5-s + (−0.461 − 2.61i)7-s + (−2.50 − 4.34i)8-s + (−3.78 + 6.55i)10-s + (−3.22 − 1.17i)11-s + (−2.56 − 2.14i)13-s + (−5.00 − 4.20i)14-s + (−3.98 − 1.45i)16-s + (1.28 − 2.22i)17-s + (1.04 + 1.81i)19-s + (2.16 + 12.2i)20-s + (−7.93 + 2.88i)22-s + (−0.0928 + 0.526i)23-s + ⋯
L(s)  = 1  + (1.33 − 1.11i)2-s + (0.350 − 1.98i)4-s + (−1.29 + 0.471i)5-s + (−0.174 − 0.989i)7-s + (−0.886 − 1.53i)8-s + (−1.19 + 2.07i)10-s + (−0.973 − 0.354i)11-s + (−0.710 − 0.596i)13-s + (−1.33 − 1.12i)14-s + (−0.997 − 0.363i)16-s + (0.311 − 0.540i)17-s + (0.240 + 0.416i)19-s + (0.483 + 2.74i)20-s + (−1.69 + 0.615i)22-s + (−0.0193 + 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.993 - 0.116i$
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: \((2,\ 729,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101686 + 1.74588i\)
\(L(\frac12)\) \(\approx\) \(0.101686 + 1.74588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 1.57i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (2.89 - 1.05i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.461 + 2.61i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (3.22 + 1.17i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (2.56 + 2.14i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.28 + 2.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.04 - 1.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0928 - 0.526i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.93 - 1.62i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.33 + 7.59i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-5.14 + 8.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.74 - 3.14i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.57 + 0.937i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.982 - 5.57i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (-1.55 + 0.566i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.49 - 14.1i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.50 + 3.77i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.40 + 12.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.940 + 1.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.1 + 11.0i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.04 - 2.55i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-2.54 - 4.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.99 - 3.63i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.46125411840791470625019045360, −9.572655116893152668871128055355, −7.71609267533037867467580444183, −7.56977392441613195673303595027, −6.07124496740267976576608218791, −5.04628112296742258748840746132, −4.13117443413267270097228363716, −3.41169542268791910393748498967, −2.56672551061248547029219776715, −0.56850399194458961795683669810, 2.65150051816232905136144995890, 3.73437984075886958836256582001, 4.76073178127113399390612332865, 5.22325766934431635416693876008, 6.36626922245150634070917916575, 7.27024643505937971576379451564, 8.002321630276743829787420268258, 8.626084987634848008761242729270, 9.869776708470417116329859575186, 11.30298683892961859479687976728

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