Properties

Label 2-3e6-1.1-c5-0-18
Degree $2$
Conductor $729$
Sign $1$
Analytic cond. $116.919$
Root an. cond. $10.8129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.98·2-s + 16.8·4-s − 77.5·5-s − 88.8·7-s − 106.·8-s − 541.·10-s − 322.·11-s − 425.·13-s − 620.·14-s − 1.27e3·16-s − 1.07e3·17-s + 1.04e3·19-s − 1.30e3·20-s − 2.25e3·22-s + 3.41e3·23-s + 2.88e3·25-s − 2.97e3·26-s − 1.49e3·28-s − 8.31e3·29-s − 9.59e3·31-s − 5.54e3·32-s − 7.51e3·34-s + 6.88e3·35-s − 1.01e3·37-s + 7.33e3·38-s + 8.22e3·40-s + 1.30e4·41-s + ⋯
L(s)  = 1  + 1.23·2-s + 0.525·4-s − 1.38·5-s − 0.684·7-s − 0.586·8-s − 1.71·10-s − 0.804·11-s − 0.697·13-s − 0.845·14-s − 1.24·16-s − 0.902·17-s + 0.666·19-s − 0.728·20-s − 0.993·22-s + 1.34·23-s + 0.922·25-s − 0.861·26-s − 0.359·28-s − 1.83·29-s − 1.79·31-s − 0.956·32-s − 1.11·34-s + 0.949·35-s − 0.121·37-s + 0.823·38-s + 0.812·40-s + 1.21·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(116.919\)
Root analytic conductor: \(10.8129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8760784932\)
\(L(\frac12)\) \(\approx\) \(0.8760784932\)
\(L(\frac{7}{2})\) 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 \( 1 - 6.98T + 32T^{2} \)
5 \( 1 + 77.5T + 3.12e3T^{2} \)
7 \( 1 + 88.8T + 1.68e4T^{2} \)
11 \( 1 + 322.T + 1.61e5T^{2} \)
13 \( 1 + 425.T + 3.71e5T^{2} \)
17 \( 1 + 1.07e3T + 1.41e6T^{2} \)
19 \( 1 - 1.04e3T + 2.47e6T^{2} \)
23 \( 1 - 3.41e3T + 6.43e6T^{2} \)
29 \( 1 + 8.31e3T + 2.05e7T^{2} \)
31 \( 1 + 9.59e3T + 2.86e7T^{2} \)
37 \( 1 + 1.01e3T + 6.93e7T^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 - 5.21e3T + 1.47e8T^{2} \)
47 \( 1 - 2.40e4T + 2.29e8T^{2} \)
53 \( 1 - 4.14e3T + 4.18e8T^{2} \)
59 \( 1 + 6.99e3T + 7.14e8T^{2} \)
61 \( 1 - 9.47e3T + 8.44e8T^{2} \)
67 \( 1 + 2.45e4T + 1.35e9T^{2} \)
71 \( 1 - 6.59e4T + 1.80e9T^{2} \)
73 \( 1 - 1.96e4T + 2.07e9T^{2} \)
79 \( 1 + 7.44e4T + 3.07e9T^{2} \)
83 \( 1 + 7.26e4T + 3.93e9T^{2} \)
89 \( 1 - 2.59e4T + 5.58e9T^{2} \)
97 \( 1 - 1.01e5T + 8.58e9T^{2} \)
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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

−9.496847041323377611257088328706, −8.842539571170459470215730470209, −7.48511961895737944230057857079, −7.13377988443172732051464412415, −5.78624516881367461911455817881, −5.01368384808253870121074888254, −4.07395004631233286005134826353, −3.38948687807783015888318074273, −2.46969052509579298972497192797, −0.33668515716549539485906006728, 0.33668515716549539485906006728, 2.46969052509579298972497192797, 3.38948687807783015888318074273, 4.07395004631233286005134826353, 5.01368384808253870121074888254, 5.78624516881367461911455817881, 7.13377988443172732051464412415, 7.48511961895737944230057857079, 8.842539571170459470215730470209, 9.496847041323377611257088328706

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