Properties

Label 2-3e6-1.1-c1-0-18
Degree $2$
Conductor $729$
Sign $1$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.40·2-s + 3.76·4-s − 0.0930·5-s − 0.579·7-s + 4.24·8-s − 0.223·10-s + 3.09·11-s + 4.20·13-s − 1.39·14-s + 2.66·16-s + 1.99·17-s − 3.84·19-s − 0.350·20-s + 7.42·22-s + 4.45·23-s − 4.99·25-s + 10.0·26-s − 2.18·28-s − 6.39·29-s + 1.65·31-s − 2.10·32-s + 4.78·34-s + 0.0539·35-s + 4.03·37-s − 9.23·38-s − 0.395·40-s + 1.09·41-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.88·4-s − 0.0416·5-s − 0.219·7-s + 1.50·8-s − 0.0706·10-s + 0.932·11-s + 1.16·13-s − 0.372·14-s + 0.665·16-s + 0.482·17-s − 0.882·19-s − 0.0784·20-s + 1.58·22-s + 0.928·23-s − 0.998·25-s + 1.97·26-s − 0.412·28-s − 1.18·29-s + 0.297·31-s − 0.371·32-s + 0.820·34-s + 0.00912·35-s + 0.662·37-s − 1.49·38-s − 0.0624·40-s + 0.171·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(4.032184741\)
\(L(\frac12)\) \(\approx\) \(4.032184741\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40T + 2T^{2} \)
5 \( 1 + 0.0930T + 5T^{2} \)
7 \( 1 + 0.579T + 7T^{2} \)
11 \( 1 - 3.09T + 11T^{2} \)
13 \( 1 - 4.20T + 13T^{2} \)
17 \( 1 - 1.99T + 17T^{2} \)
19 \( 1 + 3.84T + 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 + 6.39T + 29T^{2} \)
31 \( 1 - 1.65T + 31T^{2} \)
37 \( 1 - 4.03T + 37T^{2} \)
41 \( 1 - 1.09T + 41T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 + 3.59T + 47T^{2} \)
53 \( 1 + 5.40T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 + 8.83T + 67T^{2} \)
71 \( 1 + 1.14T + 71T^{2} \)
73 \( 1 - 0.195T + 73T^{2} \)
79 \( 1 + 7.20T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 1.55T + 89T^{2} \)
97 \( 1 - 5.29T + 97T^{2} \)
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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

−10.84883308003732015338994740309, −9.615531540883595798960937986243, −8.634197104267576725249431141829, −7.44731080190493368048680785830, −6.38468311987805840182677202037, −5.99442891828165287416195095325, −4.82369108770828682500824239274, −3.88994231302576247410678738790, −3.22355982813605523705427438329, −1.73194096264220742057972577275, 1.73194096264220742057972577275, 3.22355982813605523705427438329, 3.88994231302576247410678738790, 4.82369108770828682500824239274, 5.99442891828165287416195095325, 6.38468311987805840182677202037, 7.44731080190493368048680785830, 8.634197104267576725249431141829, 9.615531540883595798960937986243, 10.84883308003732015338994740309

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