# Properties

 Label 2-864-24.5-c2-0-16 Degree $2$ Conductor $864$ Sign $1$ Analytic cond. $23.5422$ Root an. cond. $4.85204$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.48·5-s − 13.4·7-s + 11.9·11-s + 31.0·25-s + 50·29-s + 61.4·31-s − 100.·35-s + 132.·49-s + 4.57·53-s + 89.6·55-s + 10·59-s − 143.·73-s − 161.·77-s + 58·79-s − 17.8·83-s + 128.·97-s + 154.·101-s + 10·103-s + 212.·107-s + ⋯
 L(s)  = 1 + 1.49·5-s − 1.92·7-s + 1.08·11-s + 1.24·25-s + 1.72·29-s + 1.98·31-s − 2.88·35-s + 2.71·49-s + 0.0862·53-s + 1.62·55-s + 0.169·59-s − 1.96·73-s − 2.09·77-s + 0.734·79-s − 0.215·83-s + 1.32·97-s + 1.52·101-s + 0.0970·103-s + 1.98·107-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$864$$    =    $$2^{5} \cdot 3^{3}$$ Sign: $1$ Analytic conductor: $$23.5422$$ Root analytic conductor: $$4.85204$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{864} (593, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 864,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.165699392$$ $$L(\frac12)$$ $$\approx$$ $$2.165699392$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 - 7.48T + 25T^{2}$$
7 $$1 + 13.4T + 49T^{2}$$
11 $$1 - 11.9T + 121T^{2}$$
13 $$1 - 169T^{2}$$
17 $$1 - 289T^{2}$$
19 $$1 - 361T^{2}$$
23 $$1 - 529T^{2}$$
29 $$1 - 50T + 841T^{2}$$
31 $$1 - 61.4T + 961T^{2}$$
37 $$1 - 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 - 1.84e3T^{2}$$
47 $$1 - 2.20e3T^{2}$$
53 $$1 - 4.57T + 2.80e3T^{2}$$
59 $$1 - 10T + 3.48e3T^{2}$$
61 $$1 - 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 143.T + 5.32e3T^{2}$$
79 $$1 - 58T + 6.24e3T^{2}$$
83 $$1 + 17.8T + 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 - 128.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$