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  − 4.96e4·2-s − 6.12e9·4-s − 2.04e10·5-s − 1.22e14·7-s + 7.30e14·8-s + 1.01e15·10-s − 2.15e17·11-s − 1.07e18·13-s + 6.06e18·14-s + 1.62e19·16-s − 2.54e20·17-s − 1.22e21·19-s + 1.25e20·20-s + 1.07e22·22-s + 5.11e21·23-s − 1.15e23·25-s + 5.35e22·26-s + 7.47e23·28-s + 1.64e23·29-s − 6.75e24·31-s − 7.08e24·32-s + 1.26e25·34-s + 2.49e24·35-s − 7.46e25·37-s + 6.06e25·38-s − 1.49e25·40-s − 4.96e26·41-s + ⋯
L(s)  = 1  − 0.536·2-s − 0.712·4-s − 0.0598·5-s − 1.38·7-s + 0.918·8-s + 0.0320·10-s − 1.41·11-s − 0.449·13-s + 0.744·14-s + 0.220·16-s − 1.26·17-s − 0.970·19-s + 0.0426·20-s + 0.758·22-s + 0.173·23-s − 0.996·25-s + 0.240·26-s + 0.990·28-s + 0.122·29-s − 1.66·31-s − 1.03·32-s + 0.680·34-s + 0.0831·35-s − 0.994·37-s + 0.520·38-s − 0.0549·40-s − 1.21·41-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

\(L(17)\) \(\approx\) \(0.002382050437\)
\(L(\frac12)\) \(\approx\) \(0.002382050437\)
\(L(\frac{35}{2})\) 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 \( 1 + 4.96e4T + 8.58e9T^{2} \)
5 \( 1 + 2.04e10T + 1.16e23T^{2} \)
7 \( 1 + 1.22e14T + 7.73e27T^{2} \)
11 \( 1 + 2.15e17T + 2.32e34T^{2} \)
13 \( 1 + 1.07e18T + 5.75e36T^{2} \)
17 \( 1 + 2.54e20T + 4.02e40T^{2} \)
19 \( 1 + 1.22e21T + 1.58e42T^{2} \)
23 \( 1 - 5.11e21T + 8.65e44T^{2} \)
29 \( 1 - 1.64e23T + 1.81e48T^{2} \)
31 \( 1 + 6.75e24T + 1.64e49T^{2} \)
37 \( 1 + 7.46e25T + 5.63e51T^{2} \)
41 \( 1 + 4.96e26T + 1.66e53T^{2} \)
43 \( 1 + 1.99e26T + 8.02e53T^{2} \)
47 \( 1 + 2.16e27T + 1.51e55T^{2} \)
53 \( 1 - 3.60e28T + 7.96e56T^{2} \)
59 \( 1 - 1.87e29T + 2.74e58T^{2} \)
61 \( 1 - 4.18e27T + 8.23e58T^{2} \)
67 \( 1 - 4.85e29T + 1.82e60T^{2} \)
71 \( 1 - 3.42e30T + 1.23e61T^{2} \)
73 \( 1 - 7.01e30T + 3.08e61T^{2} \)
79 \( 1 + 2.95e30T + 4.18e62T^{2} \)
83 \( 1 - 1.23e31T + 2.13e63T^{2} \)
89 \( 1 + 7.05e31T + 2.13e64T^{2} \)
97 \( 1 + 7.71e32T + 3.65e65T^{2} \)
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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

−13.46912948139148517203353618901, −12.76814587770784757622545255447, −10.62469884087428772281734725786, −9.622016537176720860979209534186, −8.411574983212849430759663194070, −6.93953776391012555694629037343, −5.26228145783937502490386112774, −3.77175510616389440508152230795, −2.21738605633794544192925859210, −0.02658994385999218551493683890, 0.02658994385999218551493683890, 2.21738605633794544192925859210, 3.77175510616389440508152230795, 5.26228145783937502490386112774, 6.93953776391012555694629037343, 8.411574983212849430759663194070, 9.622016537176720860979209534186, 10.62469884087428772281734725786, 12.76814587770784757622545255447, 13.46912948139148517203353618901

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