Properties

Label 2-3e6-27.22-c1-0-16
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  + (0.132 − 0.111i)2-s + (−0.342 + 1.94i)4-s + (3.51 − 1.27i)5-s + (0.526 + 2.98i)7-s + (0.343 + 0.594i)8-s + (0.323 − 0.559i)10-s + (2.34 + 0.852i)11-s + (−0.586 − 0.491i)13-s + (0.401 + 0.337i)14-s + (−3.59 − 1.30i)16-s + (−2.31 + 4.00i)17-s + (0.305 + 0.529i)19-s + (1.27 + 7.24i)20-s + (0.405 − 0.147i)22-s + (1.13 − 6.42i)23-s + ⋯
L(s)  = 1  + (0.0936 − 0.0786i)2-s + (−0.171 + 0.970i)4-s + (1.57 − 0.571i)5-s + (0.198 + 1.12i)7-s + (0.121 + 0.210i)8-s + (0.102 − 0.177i)10-s + (0.705 + 0.256i)11-s + (−0.162 − 0.136i)13-s + (0.107 + 0.0900i)14-s + (−0.897 − 0.326i)16-s + (−0.560 + 0.970i)17-s + (0.0701 + 0.121i)19-s + (0.285 + 1.62i)20-s + (0.0863 − 0.0314i)22-s + (0.236 − 1.33i)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: $\chi_{729} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82277 + 0.915432i\)
\(L(\frac12)\) \(\approx\) \(1.82277 + 0.915432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.132 + 0.111i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-3.51 + 1.27i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.526 - 2.98i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.34 - 0.852i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.586 + 0.491i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.31 - 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.305 - 0.529i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 + 6.42i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.01 - 4.21i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.13 + 6.45i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.47 - 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.02 - 3.38i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.23 + 1.90i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.192 - 1.09i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 + (-11.1 + 4.05i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.42 - 8.06i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.928 + 0.779i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.45 + 4.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.14 + 3.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.03 - 7.58i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.90 + 5.79i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (3.76 + 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.891 - 0.324i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.39382052283483421442609015390, −9.437109216759925918658804850432, −8.822666590475607975541423426227, −8.306700719361110820448637526172, −6.87170390037250764282678421788, −6.00422369799708670666529376059, −5.15368560289177459712530602054, −4.14434358868058666180212811249, −2.63242502792029261610946183191, −1.83126215040501387098833357912, 1.14168291110487845775563980064, 2.24090400932524626014469689624, 3.81412784179378410634975036813, 5.06189145569983154938761854857, 5.77428935519421604652656539682, 6.75966240772547210575990055015, 7.22149113100862865022404105980, 8.955749651130239692350469996318, 9.569305236109161772151919718508, 10.17831885378273593804696578214

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