# Properties

 Degree $2$ Conductor $288$ Sign $0.954 - 0.298i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.658 + 1.88i)2-s + (−3.13 + 2.48i)4-s + (0.659 − 1.59i)5-s + (9.54 − 9.54i)7-s + (−6.75 − 4.27i)8-s + (3.44 + 0.197i)10-s + (3.96 − 9.57i)11-s + (1.91 + 4.63i)13-s + (24.3 + 11.7i)14-s + (3.62 − 15.5i)16-s − 15.3i·17-s + (0.827 − 0.342i)19-s + (1.89 + 6.62i)20-s + (20.6 + 1.18i)22-s + (12.9 + 12.9i)23-s + ⋯
 L(s)  = 1 + (0.329 + 0.944i)2-s + (−0.783 + 0.621i)4-s + (0.131 − 0.318i)5-s + (1.36 − 1.36i)7-s + (−0.844 − 0.534i)8-s + (0.344 + 0.0197i)10-s + (0.360 − 0.870i)11-s + (0.147 + 0.356i)13-s + (1.73 + 0.838i)14-s + (0.226 − 0.973i)16-s − 0.900i·17-s + (0.0435 − 0.0180i)19-s + (0.0946 + 0.331i)20-s + (0.940 + 0.0538i)22-s + (0.561 + 0.561i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$288$$    =    $$2^{5} \cdot 3^{2}$$ Sign: $0.954 - 0.298i$ Motivic weight: $$2$$ Character: $\chi_{288} (235, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 288,\ (\ :1),\ 0.954 - 0.298i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.99014 + 0.303800i$$ $$L(\frac12)$$ $$\approx$$ $$1.99014 + 0.303800i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.658 - 1.88i)T$$
3 $$1$$
good5 $$1 + (-0.659 + 1.59i)T + (-17.6 - 17.6i)T^{2}$$
7 $$1 + (-9.54 + 9.54i)T - 49iT^{2}$$
11 $$1 + (-3.96 + 9.57i)T + (-85.5 - 85.5i)T^{2}$$
13 $$1 + (-1.91 - 4.63i)T + (-119. + 119. i)T^{2}$$
17 $$1 + 15.3iT - 289T^{2}$$
19 $$1 + (-0.827 + 0.342i)T + (255. - 255. i)T^{2}$$
23 $$1 + (-12.9 - 12.9i)T + 529iT^{2}$$
29 $$1 + (23.7 - 9.85i)T + (594. - 594. i)T^{2}$$
31 $$1 - 25.1iT - 961T^{2}$$
37 $$1 + (-13.6 + 32.8i)T + (-968. - 968. i)T^{2}$$
41 $$1 + (-32.9 + 32.9i)T - 1.68e3iT^{2}$$
43 $$1 + (17.9 - 43.3i)T + (-1.30e3 - 1.30e3i)T^{2}$$
47 $$1 + 20.1T + 2.20e3T^{2}$$
53 $$1 + (35.0 + 14.5i)T + (1.98e3 + 1.98e3i)T^{2}$$
59 $$1 + (-60.6 - 25.1i)T + (2.46e3 + 2.46e3i)T^{2}$$
61 $$1 + (27.9 - 11.5i)T + (2.63e3 - 2.63e3i)T^{2}$$
67 $$1 + (-1.13 - 2.73i)T + (-3.17e3 + 3.17e3i)T^{2}$$
71 $$1 + (-45.6 + 45.6i)T - 5.04e3iT^{2}$$
73 $$1 + (29.1 - 29.1i)T - 5.32e3iT^{2}$$
79 $$1 - 3.27T + 6.24e3T^{2}$$
83 $$1 + (56.7 - 23.5i)T + (4.87e3 - 4.87e3i)T^{2}$$
89 $$1 + (-44.5 - 44.5i)T + 7.92e3iT^{2}$$
97 $$1 + 106.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$