Properties

Label 2-3e6-1.1-c1-0-14
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  − 1.28·2-s − 0.347·4-s − 0.446·5-s − 3.53·7-s + 3.01·8-s + 0.573·10-s + 2.78·11-s + 3.29·13-s + 4.54·14-s − 3.18·16-s + 7.03·17-s − 5.18·19-s + 0.155·20-s − 3.57·22-s − 7.27·23-s − 4.80·25-s − 4.23·26-s + 1.22·28-s − 3.61·29-s − 1.93·31-s − 1.94·32-s − 9.04·34-s + 1.57·35-s − 3.22·37-s + 6.66·38-s − 1.34·40-s + 4.86·41-s + ⋯
L(s)  = 1  − 0.909·2-s − 0.173·4-s − 0.199·5-s − 1.33·7-s + 1.06·8-s + 0.181·10-s + 0.838·11-s + 0.912·13-s + 1.21·14-s − 0.796·16-s + 1.70·17-s − 1.18·19-s + 0.0346·20-s − 0.761·22-s − 1.51·23-s − 0.960·25-s − 0.829·26-s + 0.231·28-s − 0.672·29-s − 0.347·31-s − 0.343·32-s − 1.55·34-s + 0.266·35-s − 0.530·37-s + 1.08·38-s − 0.213·40-s + 0.759·41-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 \( 1 + 1.28T + 2T^{2} \)
5 \( 1 + 0.446T + 5T^{2} \)
7 \( 1 + 3.53T + 7T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 3.29T + 13T^{2} \)
17 \( 1 - 7.03T + 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 + 7.27T + 23T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 + 1.93T + 31T^{2} \)
37 \( 1 + 3.22T + 37T^{2} \)
41 \( 1 - 4.86T + 41T^{2} \)
43 \( 1 + 5.75T + 43T^{2} \)
47 \( 1 + 3.01T + 47T^{2} \)
53 \( 1 + 8.77T + 53T^{2} \)
59 \( 1 + 2.96T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 + 9.43T + 67T^{2} \)
71 \( 1 + 5.30T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 + 18.4T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−9.856868679589171670763357500049, −9.222386586803199988926810602389, −8.343949114742025274853597721086, −7.60268935526527204223706508839, −6.48262804058891677369048992075, −5.75522192822413679222060617235, −4.11804420121259399366825220900, −3.48319650034324299815555660693, −1.60267574387908682091371094118, 0, 1.60267574387908682091371094118, 3.48319650034324299815555660693, 4.11804420121259399366825220900, 5.75522192822413679222060617235, 6.48262804058891677369048992075, 7.60268935526527204223706508839, 8.343949114742025274853597721086, 9.222386586803199988926810602389, 9.856868679589171670763357500049

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