Properties

Degree 2
Conductor $ 3^{2} $
Sign $1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.10e4·2-s − 1.30e11·4-s + 7.16e12·5-s − 5.96e15·7-s − 2.17e16·8-s + 5.80e17·10-s − 2.06e19·11-s − 4.02e20·13-s − 4.83e20·14-s + 1.62e22·16-s + 4.55e22·17-s + 1.54e23·19-s − 9.37e23·20-s − 1.67e24·22-s − 2.42e25·23-s − 2.14e25·25-s − 3.26e25·26-s + 7.81e26·28-s − 1.52e27·29-s + 6.09e27·31-s + 4.30e27·32-s + 3.69e27·34-s − 4.27e28·35-s − 6.28e28·37-s + 1.25e28·38-s − 1.55e29·40-s + 8.11e29·41-s + ⋯
L(s)  = 1  + 0.218·2-s − 0.952·4-s + 0.840·5-s − 1.38·7-s − 0.426·8-s + 0.183·10-s − 1.11·11-s − 0.993·13-s − 0.302·14-s + 0.858·16-s + 0.786·17-s + 0.341·19-s − 0.799·20-s − 0.244·22-s − 1.55·23-s − 0.294·25-s − 0.217·26-s + 1.31·28-s − 1.34·29-s + 1.56·31-s + 0.614·32-s + 0.171·34-s − 1.16·35-s − 0.611·37-s + 0.0746·38-s − 0.358·40-s + 1.18·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9\)    =    \(3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  $\chi_{9} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 9,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $0.9093437032$
$L(\frac12)$  $\approx$  $0.9093437032$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 8.10e4T + 1.37e11T^{2} \)
5 \( 1 - 7.16e12T + 7.27e25T^{2} \)
7 \( 1 + 5.96e15T + 1.85e31T^{2} \)
11 \( 1 + 2.06e19T + 3.40e38T^{2} \)
13 \( 1 + 4.02e20T + 1.64e41T^{2} \)
17 \( 1 - 4.55e22T + 3.36e45T^{2} \)
19 \( 1 - 1.54e23T + 2.06e47T^{2} \)
23 \( 1 + 2.42e25T + 2.42e50T^{2} \)
29 \( 1 + 1.52e27T + 1.28e54T^{2} \)
31 \( 1 - 6.09e27T + 1.51e55T^{2} \)
37 \( 1 + 6.28e28T + 1.05e58T^{2} \)
41 \( 1 - 8.11e29T + 4.70e59T^{2} \)
43 \( 1 - 4.69e29T + 2.74e60T^{2} \)
47 \( 1 + 1.02e31T + 7.37e61T^{2} \)
53 \( 1 + 5.42e31T + 6.28e63T^{2} \)
59 \( 1 - 5.27e32T + 3.32e65T^{2} \)
61 \( 1 + 6.19e31T + 1.14e66T^{2} \)
67 \( 1 - 9.30e33T + 3.67e67T^{2} \)
71 \( 1 - 2.75e34T + 3.13e68T^{2} \)
73 \( 1 - 2.09e34T + 8.76e68T^{2} \)
79 \( 1 - 1.20e34T + 1.63e70T^{2} \)
83 \( 1 - 3.25e34T + 1.01e71T^{2} \)
89 \( 1 - 1.39e36T + 1.34e72T^{2} \)
97 \( 1 + 6.18e36T + 3.24e73T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.25251137523164197221312221922, −12.38626777239391465498198340505, −9.949024678308183839846740179452, −9.672804495418930106614075772370, −7.87511309095839839874909499433, −6.10880970572664590835727833554, −5.16107299421062086737074944040, −3.57770801743257322787710755178, −2.36567096958845444449601641358, −0.44934044528701019307714488010, 0.44934044528701019307714488010, 2.36567096958845444449601641358, 3.57770801743257322787710755178, 5.16107299421062086737074944040, 6.10880970572664590835727833554, 7.87511309095839839874909499433, 9.672804495418930106614075772370, 9.949024678308183839846740179452, 12.38626777239391465498198340505, 13.25251137523164197221312221922

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