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  − 3.21e5·2-s − 3.37e10·4-s − 1.53e13·5-s + 2.48e15·7-s + 5.51e16·8-s + 4.93e18·10-s + 1.32e18·11-s + 1.53e20·13-s − 8.00e20·14-s − 1.31e22·16-s − 1.04e23·17-s − 6.96e23·19-s + 5.17e23·20-s − 4.25e23·22-s − 1.11e25·23-s + 1.62e26·25-s − 4.93e25·26-s − 8.39e25·28-s − 1.43e27·29-s + 3.90e26·31-s − 3.35e27·32-s + 3.37e28·34-s − 3.81e28·35-s − 8.53e28·37-s + 2.24e29·38-s − 8.45e29·40-s − 4.51e29·41-s + ⋯
L(s)  = 1  − 0.868·2-s − 0.245·4-s − 1.79·5-s + 0.577·7-s + 1.08·8-s + 1.56·10-s + 0.0716·11-s + 0.378·13-s − 0.501·14-s − 0.694·16-s − 1.80·17-s − 1.53·19-s + 0.441·20-s − 0.0622·22-s − 0.719·23-s + 2.23·25-s − 0.328·26-s − 0.141·28-s − 1.26·29-s + 0.100·31-s − 0.479·32-s + 1.57·34-s − 1.03·35-s − 0.830·37-s + 1.33·38-s − 1.94·40-s − 0.658·41-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  =  $(2,\ 9,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $0.1334677352$
$L(\frac12)$  $\approx$  $0.1334677352$
$L(\frac{39}{2})$   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 \( 1 + 3.21e5T + 1.37e11T^{2} \)
5 \( 1 + 1.53e13T + 7.27e25T^{2} \)
7 \( 1 - 2.48e15T + 1.85e31T^{2} \)
11 \( 1 - 1.32e18T + 3.40e38T^{2} \)
13 \( 1 - 1.53e20T + 1.64e41T^{2} \)
17 \( 1 + 1.04e23T + 3.36e45T^{2} \)
19 \( 1 + 6.96e23T + 2.06e47T^{2} \)
23 \( 1 + 1.11e25T + 2.42e50T^{2} \)
29 \( 1 + 1.43e27T + 1.28e54T^{2} \)
31 \( 1 - 3.90e26T + 1.51e55T^{2} \)
37 \( 1 + 8.53e28T + 1.05e58T^{2} \)
41 \( 1 + 4.51e29T + 4.70e59T^{2} \)
43 \( 1 + 2.04e30T + 2.74e60T^{2} \)
47 \( 1 - 3.80e30T + 7.37e61T^{2} \)
53 \( 1 + 2.08e31T + 6.28e63T^{2} \)
59 \( 1 - 1.01e33T + 3.32e65T^{2} \)
61 \( 1 + 1.04e33T + 1.14e66T^{2} \)
67 \( 1 - 2.33e33T + 3.67e67T^{2} \)
71 \( 1 + 1.08e33T + 3.13e68T^{2} \)
73 \( 1 + 3.23e34T + 8.76e68T^{2} \)
79 \( 1 + 2.35e35T + 1.63e70T^{2} \)
83 \( 1 - 1.58e35T + 1.01e71T^{2} \)
89 \( 1 - 3.16e35T + 1.34e72T^{2} \)
97 \( 1 - 6.89e36T + 3.24e73T^{2} \)
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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

−13.12088307950364186850844997323, −11.54036528088754820398380295055, −10.67798314077425040803509106710, −8.757942440246284617315481444702, −8.182242227282327811052116258004, −6.95643697468195119764446931995, −4.61326015245259160371731162947, −3.87800239795090749687210611690, −1.80289714052451018836049793860, −0.21108633911225696834721495038, 0.21108633911225696834721495038, 1.80289714052451018836049793860, 3.87800239795090749687210611690, 4.61326015245259160371731162947, 6.95643697468195119764446931995, 8.182242227282327811052116258004, 8.757942440246284617315481444702, 10.67798314077425040803509106710, 11.54036528088754820398380295055, 13.12088307950364186850844997323

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