Properties

Label 2-3e2-1.1-c63-0-25
Degree $2$
Conductor $9$
Sign $-1$
Analytic cond. $226.224$
Root an. cond. $15.0407$
Motivic weight $63$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.68e9·2-s + 1.27e19·4-s + 3.68e21·5-s + 6.76e26·7-s + 1.63e28·8-s + 1.72e31·10-s + 4.60e32·11-s − 1.39e35·13-s + 3.17e36·14-s − 4.07e37·16-s − 6.98e38·17-s − 2.26e40·19-s + 4.68e40·20-s + 2.15e42·22-s − 9.79e42·23-s − 9.48e43·25-s − 6.53e44·26-s + 8.60e45·28-s − 1.02e46·29-s − 7.88e46·31-s − 3.41e47·32-s − 3.26e48·34-s + 2.49e48·35-s − 4.47e48·37-s − 1.06e50·38-s + 6.01e49·40-s + 2.61e50·41-s + ⋯
L(s)  = 1  + 1.54·2-s + 1.37·4-s + 0.353·5-s + 1.62·7-s + 0.583·8-s + 0.545·10-s + 0.723·11-s − 1.13·13-s + 2.50·14-s − 0.478·16-s − 1.21·17-s − 1.18·19-s + 0.487·20-s + 1.11·22-s − 1.24·23-s − 0.874·25-s − 1.75·26-s + 2.23·28-s − 0.880·29-s − 0.829·31-s − 1.32·32-s − 1.87·34-s + 0.573·35-s − 0.178·37-s − 1.83·38-s + 0.206·40-s + 0.411·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(64-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+63/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-1$
Analytic conductor: \(226.224\)
Root analytic conductor: \(15.0407\)
Motivic weight: \(63\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9,\ (\ :63/2),\ -1)\)

Particular Values

\(L(32)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{65}{2})\) 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 \( 1 - 4.68e9T + 9.22e18T^{2} \)
5 \( 1 - 3.68e21T + 1.08e44T^{2} \)
7 \( 1 - 6.76e26T + 1.74e53T^{2} \)
11 \( 1 - 4.60e32T + 4.05e65T^{2} \)
13 \( 1 + 1.39e35T + 1.50e70T^{2} \)
17 \( 1 + 6.98e38T + 3.29e77T^{2} \)
19 \( 1 + 2.26e40T + 3.64e80T^{2} \)
23 \( 1 + 9.79e42T + 6.14e85T^{2} \)
29 \( 1 + 1.02e46T + 1.35e92T^{2} \)
31 \( 1 + 7.88e46T + 9.03e93T^{2} \)
37 \( 1 + 4.47e48T + 6.26e98T^{2} \)
41 \( 1 - 2.61e50T + 4.03e101T^{2} \)
43 \( 1 - 1.36e51T + 8.10e102T^{2} \)
47 \( 1 - 1.83e52T + 2.19e105T^{2} \)
53 \( 1 + 3.07e54T + 4.25e108T^{2} \)
59 \( 1 + 6.07e55T + 3.66e111T^{2} \)
61 \( 1 - 2.59e56T + 2.99e112T^{2} \)
67 \( 1 + 2.81e57T + 1.10e115T^{2} \)
71 \( 1 - 2.35e58T + 4.25e116T^{2} \)
73 \( 1 - 3.66e58T + 2.45e117T^{2} \)
79 \( 1 - 9.89e59T + 3.55e119T^{2} \)
83 \( 1 + 3.72e59T + 7.97e120T^{2} \)
89 \( 1 + 2.65e61T + 6.47e122T^{2} \)
97 \( 1 - 2.06e62T + 1.46e125T^{2} \)
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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

−10.90101043348499775829730234749, −9.197568020694155968174453329393, −7.80865272464027891552597743786, −6.53835773506426198747732834909, −5.49199424530642474752849925556, −4.55069596599515836470315383789, −3.94673302739396849919545701876, −2.19513544604592946222278668656, −1.90198340580633711859173453625, 0, 1.90198340580633711859173453625, 2.19513544604592946222278668656, 3.94673302739396849919545701876, 4.55069596599515836470315383789, 5.49199424530642474752849925556, 6.53835773506426198747732834909, 7.80865272464027891552597743786, 9.197568020694155968174453329393, 10.90101043348499775829730234749

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