Properties

Label 2-3e6-1.1-c1-0-21
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  + 2.12·2-s + 2.51·4-s + 2.07·5-s + 4.84·7-s + 1.09·8-s + 4.40·10-s − 4.14·11-s − 1.21·13-s + 10.3·14-s − 2.70·16-s − 2.36·17-s − 1.83·19-s + 5.20·20-s − 8.81·22-s + 4.30·23-s − 0.711·25-s − 2.58·26-s + 12.1·28-s − 2.98·29-s + 1.47·31-s − 7.93·32-s − 5.02·34-s + 10.0·35-s + 8.97·37-s − 3.90·38-s + 2.26·40-s − 2.26·41-s + ⋯
L(s)  = 1  + 1.50·2-s + 1.25·4-s + 0.926·5-s + 1.83·7-s + 0.387·8-s + 1.39·10-s − 1.25·11-s − 0.337·13-s + 2.75·14-s − 0.675·16-s − 0.573·17-s − 0.421·19-s + 1.16·20-s − 1.87·22-s + 0.897·23-s − 0.142·25-s − 0.506·26-s + 2.30·28-s − 0.553·29-s + 0.264·31-s − 1.40·32-s − 0.861·34-s + 1.69·35-s + 1.47·37-s − 0.633·38-s + 0.358·40-s − 0.353·41-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

\(L(1)\) \(\approx\) \(4.049459232\)
\(L(\frac12)\) \(\approx\) \(4.049459232\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.12T + 2T^{2} \)
5 \( 1 - 2.07T + 5T^{2} \)
7 \( 1 - 4.84T + 7T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
17 \( 1 + 2.36T + 17T^{2} \)
19 \( 1 + 1.83T + 19T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 + 2.98T + 29T^{2} \)
31 \( 1 - 1.47T + 31T^{2} \)
37 \( 1 - 8.97T + 37T^{2} \)
41 \( 1 + 2.26T + 41T^{2} \)
43 \( 1 + 5.48T + 43T^{2} \)
47 \( 1 + 7.18T + 47T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + 0.262T + 59T^{2} \)
61 \( 1 + 4.45T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 4.55T + 79T^{2} \)
83 \( 1 - 8.45T + 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 + 5.10T + 97T^{2} \)
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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

−10.82750832380592657856312433000, −9.669767239281719153548131582813, −8.517386611967797736160404079658, −7.67391846222819376325202448823, −6.56363574793811812675857002404, −5.45891608203138155470317072219, −5.07240677257135440828915794010, −4.25940095032795052966740600149, −2.69280749121232673769795368865, −1.90226317428666647700646517728, 1.90226317428666647700646517728, 2.69280749121232673769795368865, 4.25940095032795052966740600149, 5.07240677257135440828915794010, 5.45891608203138155470317072219, 6.56363574793811812675857002404, 7.67391846222819376325202448823, 8.517386611967797736160404079658, 9.669767239281719153548131582813, 10.82750832380592657856312433000

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