# Properties

 Degree $2$ Conductor $324$ Sign $-1$ Motivic weight $5$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 110.·5-s + 101.·7-s + 150.·11-s + 635.·13-s − 1.49e3·17-s + 1.43e3·19-s + 1.26e3·23-s + 9.06e3·25-s + 2.77e3·29-s − 6.96e3·31-s − 1.12e4·35-s − 7.95e3·37-s + 2.02e3·41-s − 1.25e4·43-s + 6.48e3·47-s − 6.45e3·49-s − 9.82e3·53-s − 1.65e4·55-s − 4.70e4·59-s + 8.33e3·61-s − 7.01e4·65-s − 7.26e3·67-s − 3.58e3·71-s + 5.80e4·73-s + 1.52e4·77-s − 6.37e4·79-s − 8.28e4·83-s + ⋯
 L(s)  = 1 − 1.97·5-s + 0.784·7-s + 0.374·11-s + 1.04·13-s − 1.25·17-s + 0.913·19-s + 0.498·23-s + 2.90·25-s + 0.613·29-s − 1.30·31-s − 1.54·35-s − 0.954·37-s + 0.188·41-s − 1.03·43-s + 0.428·47-s − 0.384·49-s − 0.480·53-s − 0.739·55-s − 1.76·59-s + 0.286·61-s − 2.05·65-s − 0.197·67-s − 0.0843·71-s + 1.27·73-s + 0.293·77-s − 1.14·79-s − 1.32·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $-1$ Motivic weight: $$5$$ Character: $\chi_{324} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 324,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 110.T + 3.12e3T^{2}$$
7 $$1 - 101.T + 1.68e4T^{2}$$
11 $$1 - 150.T + 1.61e5T^{2}$$
13 $$1 - 635.T + 3.71e5T^{2}$$
17 $$1 + 1.49e3T + 1.41e6T^{2}$$
19 $$1 - 1.43e3T + 2.47e6T^{2}$$
23 $$1 - 1.26e3T + 6.43e6T^{2}$$
29 $$1 - 2.77e3T + 2.05e7T^{2}$$
31 $$1 + 6.96e3T + 2.86e7T^{2}$$
37 $$1 + 7.95e3T + 6.93e7T^{2}$$
41 $$1 - 2.02e3T + 1.15e8T^{2}$$
43 $$1 + 1.25e4T + 1.47e8T^{2}$$
47 $$1 - 6.48e3T + 2.29e8T^{2}$$
53 $$1 + 9.82e3T + 4.18e8T^{2}$$
59 $$1 + 4.70e4T + 7.14e8T^{2}$$
61 $$1 - 8.33e3T + 8.44e8T^{2}$$
67 $$1 + 7.26e3T + 1.35e9T^{2}$$
71 $$1 + 3.58e3T + 1.80e9T^{2}$$
73 $$1 - 5.80e4T + 2.07e9T^{2}$$
79 $$1 + 6.37e4T + 3.07e9T^{2}$$
83 $$1 + 8.28e4T + 3.93e9T^{2}$$
89 $$1 - 3.86e3T + 5.58e9T^{2}$$
97 $$1 - 6.92e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$