Properties

Label 2-3e6-1.1-c1-0-2
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  + 0.172·2-s − 1.97·4-s − 3.73·5-s + 3.03·7-s − 0.686·8-s − 0.646·10-s − 2.49·11-s − 0.765·13-s + 0.524·14-s + 3.82·16-s + 4.62·17-s − 0.611·19-s + 7.36·20-s − 0.431·22-s + 6.52·23-s + 8.96·25-s − 0.132·26-s − 5.97·28-s − 6.55·29-s + 6.55·31-s + 2.03·32-s + 0.799·34-s − 11.3·35-s + 4.95·37-s − 0.105·38-s + 2.56·40-s + 5.26·41-s + ⋯
L(s)  = 1  + 0.122·2-s − 0.985·4-s − 1.67·5-s + 1.14·7-s − 0.242·8-s − 0.204·10-s − 0.751·11-s − 0.212·13-s + 0.140·14-s + 0.955·16-s + 1.12·17-s − 0.140·19-s + 1.64·20-s − 0.0918·22-s + 1.36·23-s + 1.79·25-s − 0.0259·26-s − 1.12·28-s − 1.21·29-s + 1.17·31-s + 0.359·32-s + 0.137·34-s − 1.91·35-s + 0.815·37-s − 0.0171·38-s + 0.405·40-s + 0.821·41-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

\(L(1)\) \(\approx\) \(0.9627861164\)
\(L(\frac12)\) \(\approx\) \(0.9627861164\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.172T + 2T^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 - 3.03T + 7T^{2} \)
11 \( 1 + 2.49T + 11T^{2} \)
13 \( 1 + 0.765T + 13T^{2} \)
17 \( 1 - 4.62T + 17T^{2} \)
19 \( 1 + 0.611T + 19T^{2} \)
23 \( 1 - 6.52T + 23T^{2} \)
29 \( 1 + 6.55T + 29T^{2} \)
31 \( 1 - 6.55T + 31T^{2} \)
37 \( 1 - 4.95T + 37T^{2} \)
41 \( 1 - 5.26T + 41T^{2} \)
43 \( 1 - 5.57T + 43T^{2} \)
47 \( 1 - 1.10T + 47T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 8.18T + 61T^{2} \)
67 \( 1 + 1.21T + 67T^{2} \)
71 \( 1 + 4.91T + 71T^{2} \)
73 \( 1 - 4.29T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 9.01T + 83T^{2} \)
89 \( 1 - 7.53T + 89T^{2} \)
97 \( 1 + 0.948T + 97T^{2} \)
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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

−10.57484870106314574914296963722, −9.399411302239532608072998550108, −8.441906588099580354933921432222, −7.86624929305702151148931412361, −7.35368398273338432605387539775, −5.58476695851257777719103905831, −4.76568339855758623637334294215, −4.10284203300566520078979411695, −3.01302537208609964447957061125, −0.829257742670493852461042322898, 0.829257742670493852461042322898, 3.01302537208609964447957061125, 4.10284203300566520078979411695, 4.76568339855758623637334294215, 5.58476695851257777719103905831, 7.35368398273338432605387539775, 7.86624929305702151148931412361, 8.441906588099580354933921432222, 9.399411302239532608072998550108, 10.57484870106314574914296963722

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