Properties

Label 2-3e6-1.1-c1-0-28
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  + 1.68·2-s + 0.825·4-s − 1.12·5-s − 3.90·7-s − 1.97·8-s − 1.89·10-s − 1.87·11-s − 0.732·13-s − 6.57·14-s − 4.96·16-s + 1.88·17-s + 2.74·19-s − 0.932·20-s − 3.14·22-s − 5.82·23-s − 3.72·25-s − 1.23·26-s − 3.22·28-s − 5.31·29-s + 1.34·31-s − 4.40·32-s + 3.17·34-s + 4.41·35-s + 3.39·37-s + 4.61·38-s + 2.23·40-s + 1.79·41-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.412·4-s − 0.505·5-s − 1.47·7-s − 0.698·8-s − 0.600·10-s − 0.563·11-s − 0.203·13-s − 1.75·14-s − 1.24·16-s + 0.458·17-s + 0.629·19-s − 0.208·20-s − 0.670·22-s − 1.21·23-s − 0.744·25-s − 0.241·26-s − 0.609·28-s − 0.987·29-s + 0.240·31-s − 0.778·32-s + 0.544·34-s + 0.746·35-s + 0.558·37-s + 0.747·38-s + 0.352·40-s + 0.280·41-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 \( 1 - 1.68T + 2T^{2} \)
5 \( 1 + 1.12T + 5T^{2} \)
7 \( 1 + 3.90T + 7T^{2} \)
11 \( 1 + 1.87T + 11T^{2} \)
13 \( 1 + 0.732T + 13T^{2} \)
17 \( 1 - 1.88T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + 5.82T + 23T^{2} \)
29 \( 1 + 5.31T + 29T^{2} \)
31 \( 1 - 1.34T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 - 5.03T + 43T^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 5.23T + 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 9.88T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 5.72T + 89T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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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

−9.839206657547177039656060147527, −9.370643373934861421901235159245, −8.065194869266778125089429040670, −7.16177997306440306943283446998, −6.08290033506018708127006184561, −5.53440045323573199354654852474, −4.24543471662823034700266946181, −3.52951825264974116409177435744, −2.62881899175052538848358130059, 0, 2.62881899175052538848358130059, 3.52951825264974116409177435744, 4.24543471662823034700266946181, 5.53440045323573199354654852474, 6.08290033506018708127006184561, 7.16177997306440306943283446998, 8.065194869266778125089429040670, 9.370643373934861421901235159245, 9.839206657547177039656060147527

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