# Properties

 Label 2-18e2-4.3-c2-0-27 Degree $2$ Conductor $324$ Sign $1$ Analytic cond. $8.82836$ Root an. cond. $2.97125$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4·4-s − 1.19·5-s + 8·8-s − 2.39·10-s + 25.7·13-s + 16·16-s + 17.9·17-s − 4.78·20-s − 23.5·25-s + 51.5·26-s − 56.3·29-s + 32·32-s + 35.9·34-s + 55.7·37-s − 9.56·40-s − 80·41-s + 49·49-s − 47.1·50-s + 103.·52-s − 56·53-s − 112.·58-s − 92.9·61-s + 64·64-s − 30.8·65-s + 71.9·68-s + 28.1·73-s + ⋯
 L(s)  = 1 + 2-s + 4-s − 0.239·5-s + 8-s − 0.239·10-s + 1.98·13-s + 16-s + 1.05·17-s − 0.239·20-s − 0.942·25-s + 1.98·26-s − 1.94·29-s + 32-s + 1.05·34-s + 1.50·37-s − 0.239·40-s − 1.95·41-s + 0.999·49-s − 0.942·50-s + 1.98·52-s − 1.05·53-s − 1.94·58-s − 1.52·61-s + 64-s − 0.474·65-s + 1.05·68-s + 0.385·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$8.82836$$ Root analytic conductor: $$2.97125$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{324} (163, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 324,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.225963172$$ $$L(\frac12)$$ $$\approx$$ $$3.225963172$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 2T$$
3 $$1$$
good5 $$1 + 1.19T + 25T^{2}$$
7 $$1 - 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 - 25.7T + 169T^{2}$$
17 $$1 - 17.9T + 289T^{2}$$
19 $$1 - 361T^{2}$$
23 $$1 - 529T^{2}$$
29 $$1 + 56.3T + 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 - 55.7T + 1.36e3T^{2}$$
41 $$1 + 80T + 1.68e3T^{2}$$
43 $$1 - 1.84e3T^{2}$$
47 $$1 - 2.20e3T^{2}$$
53 $$1 + 56T + 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 + 92.9T + 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 28.1T + 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 - 6.88e3T^{2}$$
89 $$1 + 147.T + 7.92e3T^{2}$$
97 $$1 + 130T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$