Properties

Label 2-3e6-1.1-c1-0-17
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.684·2-s − 1.53·4-s − 1.04·5-s − 0.120·7-s + 2.41·8-s + 0.716·10-s + 5.43·11-s − 4.57·13-s + 0.0825·14-s + 1.41·16-s − 4.77·17-s − 0.588·19-s + 1.60·20-s − 3.71·22-s + 7.79·23-s − 3.90·25-s + 3.12·26-s + 0.184·28-s − 5.06·29-s − 8.75·31-s − 5.79·32-s + 3.26·34-s + 0.126·35-s − 2.18·37-s + 0.402·38-s − 2.53·40-s − 7.55·41-s + ⋯
L(s)  = 1  − 0.483·2-s − 0.766·4-s − 0.468·5-s − 0.0455·7-s + 0.854·8-s + 0.226·10-s + 1.63·11-s − 1.26·13-s + 0.0220·14-s + 0.352·16-s − 1.15·17-s − 0.135·19-s + 0.359·20-s − 0.792·22-s + 1.62·23-s − 0.780·25-s + 0.613·26-s + 0.0349·28-s − 0.941·29-s − 1.57·31-s − 1.02·32-s + 0.560·34-s + 0.0213·35-s − 0.359·37-s + 0.0653·38-s − 0.400·40-s − 1.18·41-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 \( 1 + 0.684T + 2T^{2} \)
5 \( 1 + 1.04T + 5T^{2} \)
7 \( 1 + 0.120T + 7T^{2} \)
11 \( 1 - 5.43T + 11T^{2} \)
13 \( 1 + 4.57T + 13T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 0.588T + 19T^{2} \)
23 \( 1 - 7.79T + 23T^{2} \)
29 \( 1 + 5.06T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
37 \( 1 + 2.18T + 37T^{2} \)
41 \( 1 + 7.55T + 41T^{2} \)
43 \( 1 + 1.30T + 43T^{2} \)
47 \( 1 + 2.41T + 47T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 - 0.0439T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + 1.85T + 67T^{2} \)
71 \( 1 - 6.51T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 0.702T + 79T^{2} \)
83 \( 1 + 6.77T + 83T^{2} \)
89 \( 1 + 6.85T + 89T^{2} \)
97 \( 1 + 9.02T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.575543371789038722360338385536, −9.254689257133067629298870304930, −8.438753577590634662886775143294, −7.35772262877906638168554246016, −6.73923831997645462735296113790, −5.27003126035738856807944713383, −4.39161919178226468263374453713, −3.52835046676554308811434274525, −1.72010988515115534229040122010, 0, 1.72010988515115534229040122010, 3.52835046676554308811434274525, 4.39161919178226468263374453713, 5.27003126035738856807944713383, 6.73923831997645462735296113790, 7.35772262877906638168554246016, 8.438753577590634662886775143294, 9.254689257133067629298870304930, 9.575543371789038722360338385536

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