Properties

Label 2-3e6-1.1-c1-0-8
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  − 1.57·2-s + 0.491·4-s + 1.67·5-s + 2.77·7-s + 2.38·8-s − 2.64·10-s − 4.15·11-s + 6.87·13-s − 4.38·14-s − 4.74·16-s + 0.976·17-s + 2.68·19-s + 0.824·20-s + 6.55·22-s + 1.61·23-s − 2.18·25-s − 10.8·26-s + 1.36·28-s − 8.22·29-s + 1.04·31-s + 2.72·32-s − 1.54·34-s + 4.66·35-s − 1.30·37-s − 4.23·38-s + 3.99·40-s − 4.84·41-s + ⋯
L(s)  = 1  − 1.11·2-s + 0.245·4-s + 0.750·5-s + 1.05·7-s + 0.841·8-s − 0.837·10-s − 1.25·11-s + 1.90·13-s − 1.17·14-s − 1.18·16-s + 0.236·17-s + 0.616·19-s + 0.184·20-s + 1.39·22-s + 0.336·23-s − 0.436·25-s − 2.12·26-s + 0.258·28-s − 1.52·29-s + 0.187·31-s + 0.481·32-s − 0.264·34-s + 0.788·35-s − 0.215·37-s − 0.687·38-s + 0.631·40-s − 0.757·41-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

\(L(1)\) \(\approx\) \(1.059222498\)
\(L(\frac12)\) \(\approx\) \(1.059222498\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.57T + 2T^{2} \)
5 \( 1 - 1.67T + 5T^{2} \)
7 \( 1 - 2.77T + 7T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 - 6.87T + 13T^{2} \)
17 \( 1 - 0.976T + 17T^{2} \)
19 \( 1 - 2.68T + 19T^{2} \)
23 \( 1 - 1.61T + 23T^{2} \)
29 \( 1 + 8.22T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
37 \( 1 + 1.30T + 37T^{2} \)
41 \( 1 + 4.84T + 41T^{2} \)
43 \( 1 - 9.84T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 - 9.05T + 59T^{2} \)
61 \( 1 + 1.28T + 61T^{2} \)
67 \( 1 + 4.64T + 67T^{2} \)
71 \( 1 - 5.62T + 71T^{2} \)
73 \( 1 + 4.56T + 73T^{2} \)
79 \( 1 - 4.65T + 79T^{2} \)
83 \( 1 - 5.76T + 83T^{2} \)
89 \( 1 - 4.54T + 89T^{2} \)
97 \( 1 - 8.57T + 97T^{2} \)
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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

−10.45932299482688333501306776339, −9.411206781486181995646817152710, −8.719342320152038539880779646738, −7.983455581894657076721036042878, −7.31838791955465934951377609864, −5.86091592589307152553643596876, −5.19378731119956203054696392547, −3.87763835950422610210172208167, −2.17405774737837463375929275576, −1.11744055592954130864401262748, 1.11744055592954130864401262748, 2.17405774737837463375929275576, 3.87763835950422610210172208167, 5.19378731119956203054696392547, 5.86091592589307152553643596876, 7.31838791955465934951377609864, 7.983455581894657076721036042878, 8.719342320152038539880779646738, 9.411206781486181995646817152710, 10.45932299482688333501306776339

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