Properties

Label 2-3e6-1.1-c1-0-24
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  + 2.45·2-s + 4.04·4-s + 3.08·5-s − 2.65·7-s + 5.01·8-s + 7.57·10-s + 3.43·11-s − 3.34·13-s − 6.53·14-s + 4.24·16-s − 2.57·17-s − 2.09·19-s + 12.4·20-s + 8.44·22-s − 0.534·23-s + 4.48·25-s − 8.21·26-s − 10.7·28-s − 2.53·29-s + 7.71·31-s + 0.404·32-s − 6.32·34-s − 8.18·35-s − 10.2·37-s − 5.15·38-s + 15.4·40-s + 4.88·41-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.02·4-s + 1.37·5-s − 1.00·7-s + 1.77·8-s + 2.39·10-s + 1.03·11-s − 0.927·13-s − 1.74·14-s + 1.06·16-s − 0.623·17-s − 0.481·19-s + 2.78·20-s + 1.79·22-s − 0.111·23-s + 0.897·25-s − 1.61·26-s − 2.03·28-s − 0.469·29-s + 1.38·31-s + 0.0715·32-s − 1.08·34-s − 1.38·35-s − 1.69·37-s − 0.835·38-s + 2.44·40-s + 0.762·41-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

\(L(1)\) \(\approx\) \(4.509235119\)
\(L(\frac12)\) \(\approx\) \(4.509235119\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.45T + 2T^{2} \)
5 \( 1 - 3.08T + 5T^{2} \)
7 \( 1 + 2.65T + 7T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + 3.34T + 13T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 + 2.09T + 19T^{2} \)
23 \( 1 + 0.534T + 23T^{2} \)
29 \( 1 + 2.53T + 29T^{2} \)
31 \( 1 - 7.71T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 - 2.74T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 + 5.87T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 1.88T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + 3.96T + 83T^{2} \)
89 \( 1 + 5.09T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−10.45993401100749620352636241800, −9.701782845455050464096387885269, −8.912867490282402835491836840995, −7.14905099169695093529453024671, −6.44919069617124175684135911984, −5.95935953697554333580735664585, −4.96916465618776219066087924016, −4.00043091038429119732600511879, −2.86788501803561593153001593501, −1.97151419509781647553660511517, 1.97151419509781647553660511517, 2.86788501803561593153001593501, 4.00043091038429119732600511879, 4.96916465618776219066087924016, 5.95935953697554333580735664585, 6.44919069617124175684135911984, 7.14905099169695093529453024671, 8.912867490282402835491836840995, 9.701782845455050464096387885269, 10.45993401100749620352636241800

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