Properties

Degree $2$
Conductor $9$
Sign $1$
Motivic weight $33$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.71e5·2-s + 2.07e10·4-s + 2.01e11·5-s + 5.50e13·7-s + 2.08e15·8-s + 3.45e16·10-s + 8.18e16·11-s − 1.90e18·13-s + 9.42e18·14-s + 1.79e20·16-s + 3.33e20·17-s − 1.40e20·19-s + 4.18e21·20-s + 1.40e22·22-s − 3.12e22·23-s − 7.58e22·25-s − 3.26e23·26-s + 1.14e24·28-s + 1.50e24·29-s + 5.18e23·31-s + 1.27e25·32-s + 5.72e25·34-s + 1.10e25·35-s − 3.01e25·37-s − 2.40e25·38-s + 4.20e26·40-s + 2.18e26·41-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.41·4-s + 0.590·5-s + 0.625·7-s + 2.62·8-s + 1.09·10-s + 0.537·11-s − 0.793·13-s + 1.15·14-s + 2.43·16-s + 1.66·17-s − 0.111·19-s + 1.42·20-s + 0.993·22-s − 1.06·23-s − 0.651·25-s − 1.46·26-s + 1.51·28-s + 1.12·29-s + 0.128·31-s + 1.87·32-s + 3.07·34-s + 0.369·35-s − 0.401·37-s − 0.206·38-s + 1.54·40-s + 0.536·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Motivic weight: \(33\)
Character: $\chi_{9} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :33/2),\ 1)\)

Particular Values

\(L(17)\) \(\approx\) \(8.906251616\)
\(L(\frac12)\) \(\approx\) \(8.906251616\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.71e5T + 8.58e9T^{2} \)
5 \( 1 - 2.01e11T + 1.16e23T^{2} \)
7 \( 1 - 5.50e13T + 7.73e27T^{2} \)
11 \( 1 - 8.18e16T + 2.32e34T^{2} \)
13 \( 1 + 1.90e18T + 5.75e36T^{2} \)
17 \( 1 - 3.33e20T + 4.02e40T^{2} \)
19 \( 1 + 1.40e20T + 1.58e42T^{2} \)
23 \( 1 + 3.12e22T + 8.65e44T^{2} \)
29 \( 1 - 1.50e24T + 1.81e48T^{2} \)
31 \( 1 - 5.18e23T + 1.64e49T^{2} \)
37 \( 1 + 3.01e25T + 5.63e51T^{2} \)
41 \( 1 - 2.18e26T + 1.66e53T^{2} \)
43 \( 1 - 1.76e27T + 8.02e53T^{2} \)
47 \( 1 + 3.25e27T + 1.51e55T^{2} \)
53 \( 1 + 9.17e27T + 7.96e56T^{2} \)
59 \( 1 - 1.18e29T + 2.74e58T^{2} \)
61 \( 1 + 9.92e27T + 8.23e58T^{2} \)
67 \( 1 - 1.11e30T + 1.82e60T^{2} \)
71 \( 1 + 7.58e29T + 1.23e61T^{2} \)
73 \( 1 + 6.06e30T + 3.08e61T^{2} \)
79 \( 1 + 5.57e30T + 4.18e62T^{2} \)
83 \( 1 + 4.13e31T + 2.13e63T^{2} \)
89 \( 1 + 6.21e31T + 2.13e64T^{2} \)
97 \( 1 - 4.04e32T + 3.65e65T^{2} \)
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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

−14.08015330685980784913447413397, −12.54422226576142171013309887400, −11.65882201042686300120446471056, −10.07289973263252693515985927038, −7.66309016926825645684712359703, −6.18307145153677745740702837569, −5.19962423923673568072910494536, −4.00795900289828917689815351764, −2.62532772055263355972238419670, −1.46123154365470444616523474221, 1.46123154365470444616523474221, 2.62532772055263355972238419670, 4.00795900289828917689815351764, 5.19962423923673568072910494536, 6.18307145153677745740702837569, 7.66309016926825645684712359703, 10.07289973263252693515985927038, 11.65882201042686300120446471056, 12.54422226576142171013309887400, 14.08015330685980784913447413397

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