# Properties

 Label 2-288-8.5-c7-0-32 Degree $2$ Conductor $288$ Sign $-0.857 - 0.513i$ Analytic cond. $89.9668$ Root an. cond. $9.48508$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 324. i·5-s + 956.·7-s + 5.45e3i·11-s − 6.28e3i·13-s − 3.45e4·17-s + 1.45e4i·19-s − 2.46e4·23-s − 2.71e4·25-s − 1.71e5i·29-s − 1.11e5·31-s − 3.10e5i·35-s − 1.03e5i·37-s − 7.16e4·41-s + 3.28e5i·43-s + 1.19e5·47-s + ⋯
 L(s)  = 1 − 1.16i·5-s + 1.05·7-s + 1.23i·11-s − 0.793i·13-s − 1.70·17-s + 0.488i·19-s − 0.422·23-s − 0.347·25-s − 1.30i·29-s − 0.673·31-s − 1.22i·35-s − 0.336i·37-s − 0.162·41-s + 0.629i·43-s + 0.167·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$288$$    =    $$2^{5} \cdot 3^{2}$$ Sign: $-0.857 - 0.513i$ Analytic conductor: $$89.9668$$ Root analytic conductor: $$9.48508$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{288} (145, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 288,\ (\ :7/2),\ -0.857 - 0.513i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$0.03319267175$$ $$L(\frac12)$$ $$\approx$$ $$0.03319267175$$ $$L(\frac{9}{2})$$ 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 + 324. iT - 7.81e4T^{2}$$
7 $$1 - 956.T + 8.23e5T^{2}$$
11 $$1 - 5.45e3iT - 1.94e7T^{2}$$
13 $$1 + 6.28e3iT - 6.27e7T^{2}$$
17 $$1 + 3.45e4T + 4.10e8T^{2}$$
19 $$1 - 1.45e4iT - 8.93e8T^{2}$$
23 $$1 + 2.46e4T + 3.40e9T^{2}$$
29 $$1 + 1.71e5iT - 1.72e10T^{2}$$
31 $$1 + 1.11e5T + 2.75e10T^{2}$$
37 $$1 + 1.03e5iT - 9.49e10T^{2}$$
41 $$1 + 7.16e4T + 1.94e11T^{2}$$
43 $$1 - 3.28e5iT - 2.71e11T^{2}$$
47 $$1 - 1.19e5T + 5.06e11T^{2}$$
53 $$1 - 1.04e6iT - 1.17e12T^{2}$$
59 $$1 + 2.25e5iT - 2.48e12T^{2}$$
61 $$1 - 1.55e6iT - 3.14e12T^{2}$$
67 $$1 + 3.16e5iT - 6.06e12T^{2}$$
71 $$1 - 5.38e5T + 9.09e12T^{2}$$
73 $$1 + 2.68e6T + 1.10e13T^{2}$$
79 $$1 + 8.22e6T + 1.92e13T^{2}$$
83 $$1 + 5.89e6iT - 2.71e13T^{2}$$
89 $$1 + 4.37e5T + 4.42e13T^{2}$$
97 $$1 + 7.84e6T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$