Properties

Degree 2
Conductor $ 3^{2} $
Sign $1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 50.1·2-s + 471.·4-s − 1.12e4·5-s + 5.81e4·7-s + 7.91e4·8-s + 5.64e5·10-s + 1.60e5·11-s + 7.62e5·13-s − 2.91e6·14-s − 4.93e6·16-s + 8.97e6·17-s − 1.03e7·19-s − 5.30e6·20-s − 8.06e6·22-s − 1.03e7·23-s + 7.76e7·25-s − 3.82e7·26-s + 2.74e7·28-s + 1.41e8·29-s + 1.06e8·31-s + 8.58e7·32-s − 4.50e8·34-s − 6.53e8·35-s − 9.57e6·37-s + 5.17e8·38-s − 8.89e8·40-s + 1.08e8·41-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.230·4-s − 1.60·5-s + 1.30·7-s + 0.853·8-s + 1.78·10-s + 0.300·11-s + 0.569·13-s − 1.44·14-s − 1.17·16-s + 1.53·17-s − 0.954·19-s − 0.370·20-s − 0.333·22-s − 0.336·23-s + 1.58·25-s − 0.631·26-s + 0.301·28-s + 1.27·29-s + 0.665·31-s + 0.452·32-s − 1.70·34-s − 2.10·35-s − 0.0226·37-s + 1.05·38-s − 1.37·40-s + 0.146·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9\)    =    \(3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{9} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 9,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $0.742177$
$L(\frac12)$  $\approx$  $0.742177$
$L(\frac{13}{2})$   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 \( 1 + 50.1T + 2.04e3T^{2} \)
5 \( 1 + 1.12e4T + 4.88e7T^{2} \)
7 \( 1 - 5.81e4T + 1.97e9T^{2} \)
11 \( 1 - 1.60e5T + 2.85e11T^{2} \)
13 \( 1 - 7.62e5T + 1.79e12T^{2} \)
17 \( 1 - 8.97e6T + 3.42e13T^{2} \)
19 \( 1 + 1.03e7T + 1.16e14T^{2} \)
23 \( 1 + 1.03e7T + 9.52e14T^{2} \)
29 \( 1 - 1.41e8T + 1.22e16T^{2} \)
31 \( 1 - 1.06e8T + 2.54e16T^{2} \)
37 \( 1 + 9.57e6T + 1.77e17T^{2} \)
41 \( 1 - 1.08e8T + 5.50e17T^{2} \)
43 \( 1 - 1.59e9T + 9.29e17T^{2} \)
47 \( 1 - 1.44e9T + 2.47e18T^{2} \)
53 \( 1 + 1.05e9T + 9.26e18T^{2} \)
59 \( 1 - 5.77e9T + 3.01e19T^{2} \)
61 \( 1 + 3.09e9T + 4.35e19T^{2} \)
67 \( 1 + 9.11e9T + 1.22e20T^{2} \)
71 \( 1 + 3.35e9T + 2.31e20T^{2} \)
73 \( 1 - 6.20e8T + 3.13e20T^{2} \)
79 \( 1 - 1.06e10T + 7.47e20T^{2} \)
83 \( 1 + 6.00e10T + 1.28e21T^{2} \)
89 \( 1 - 6.14e10T + 2.77e21T^{2} \)
97 \( 1 - 1.31e11T + 7.15e21T^{2} \)
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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

−18.67343877165125944631532922126, −17.32589479212792971417885542288, −15.95126040847141031770763697971, −14.41189229539374656407154591115, −11.95026666745887099259913261182, −10.69373083939767882024708034014, −8.497689203601718893741955119524, −7.69143782419466166091328575633, −4.30961620486152328390978910793, −0.952851320315113119551955304117, 0.952851320315113119551955304117, 4.30961620486152328390978910793, 7.69143782419466166091328575633, 8.497689203601718893741955119524, 10.69373083939767882024708034014, 11.95026666745887099259913261182, 14.41189229539374656407154591115, 15.95126040847141031770763697971, 17.32589479212792971417885542288, 18.67343877165125944631532922126

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