# Properties

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

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## 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 = $(2,\ 9,\ (\ :11/2),\ 1)$ $L(6)$ $\approx$ $3.25731$ $L(\frac12)$ $\approx$ $3.25731$ $L(\frac{13}{2})$ 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

−18.19580992085115734161371859212, −17.39380097503897935322562441146, −15.06912800550075981930739313953, −13.90728196048111842129494781100, −13.02984707344792266056126019917, −10.99426744886364717451432122583, −8.944512953201804008681975658185, −6.08849139898262725735604198615, −4.71439852597627973514333471125, −2.09580760927097412764734178325, 2.09580760927097412764734178325, 4.71439852597627973514333471125, 6.08849139898262725735604198615, 8.944512953201804008681975658185, 10.99426744886364717451432122583, 13.02984707344792266056126019917, 13.90728196048111842129494781100, 15.06912800550075981930739313953, 17.39380097503897935322562441146, 18.19580992085115734161371859212