Properties

Label 2-3e6-1.1-c1-0-26
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.05·2-s − 0.888·4-s − 1.74·5-s + 2.45·7-s − 3.04·8-s − 1.84·10-s − 1.25·11-s − 4.54·13-s + 2.59·14-s − 1.43·16-s − 6.64·17-s + 0.249·19-s + 1.55·20-s − 1.32·22-s + 0.842·23-s − 1.95·25-s − 4.79·26-s − 2.18·28-s − 0.512·29-s − 0.820·31-s + 4.57·32-s − 7.00·34-s − 4.29·35-s + 2.60·37-s + 0.262·38-s + 5.31·40-s − 8.15·41-s + ⋯
L(s)  = 1  + 0.745·2-s − 0.444·4-s − 0.780·5-s + 0.929·7-s − 1.07·8-s − 0.582·10-s − 0.378·11-s − 1.26·13-s + 0.692·14-s − 0.358·16-s − 1.61·17-s + 0.0571·19-s + 0.346·20-s − 0.281·22-s + 0.175·23-s − 0.390·25-s − 0.940·26-s − 0.412·28-s − 0.0951·29-s − 0.147·31-s + 0.809·32-s − 1.20·34-s − 0.725·35-s + 0.428·37-s + 0.0426·38-s + 0.840·40-s − 1.27·41-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: \(1\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.05T + 2T^{2} \)
5 \( 1 + 1.74T + 5T^{2} \)
7 \( 1 - 2.45T + 7T^{2} \)
11 \( 1 + 1.25T + 11T^{2} \)
13 \( 1 + 4.54T + 13T^{2} \)
17 \( 1 + 6.64T + 17T^{2} \)
19 \( 1 - 0.249T + 19T^{2} \)
23 \( 1 - 0.842T + 23T^{2} \)
29 \( 1 + 0.512T + 29T^{2} \)
31 \( 1 + 0.820T + 31T^{2} \)
37 \( 1 - 2.60T + 37T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + 5.30T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 3.00T + 59T^{2} \)
61 \( 1 - 2.88T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 0.0894T + 71T^{2} \)
73 \( 1 + 5.32T + 73T^{2} \)
79 \( 1 - 4.77T + 79T^{2} \)
83 \( 1 - 8.04T + 83T^{2} \)
89 \( 1 - 6.70T + 89T^{2} \)
97 \( 1 - 5.49T + 97T^{2} \)
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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

−9.951723136798602990986895106075, −9.015895125244081630811074401617, −8.186795186788081279533528059026, −7.42935156140686612403097709012, −6.29582690604307505329882815995, −4.93136339462419425007243312248, −4.70689890497037420915147465874, −3.58707604508317782956021549136, −2.27617849466440400491199217303, 0, 2.27617849466440400491199217303, 3.58707604508317782956021549136, 4.70689890497037420915147465874, 4.93136339462419425007243312248, 6.29582690604307505329882815995, 7.42935156140686612403097709012, 8.186795186788081279533528059026, 9.015895125244081630811074401617, 9.951723136798602990986895106075

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