# Properties

 Label 2-48-48.5-c2-0-7 Degree $2$ Conductor $48$ Sign $0.942 - 0.333i$ Analytic cond. $1.30790$ Root an. cond. $1.14363$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.96 − 0.391i)2-s + (−0.164 + 2.99i)3-s + (3.69 − 1.53i)4-s + (−3.61 + 3.61i)5-s + (0.848 + 5.93i)6-s − 12.2i·7-s + (6.64 − 4.45i)8-s + (−8.94 − 0.985i)9-s + (−5.67 + 8.49i)10-s + (1.76 − 1.76i)11-s + (3.98 + 11.3i)12-s + (−2.38 + 2.38i)13-s + (−4.80 − 24.0i)14-s + (−10.2 − 11.4i)15-s + (11.2 − 11.3i)16-s + 20.0i·17-s + ⋯
 L(s)  = 1 + (0.980 − 0.195i)2-s + (−0.0548 + 0.998i)3-s + (0.923 − 0.383i)4-s + (−0.722 + 0.722i)5-s + (0.141 + 0.989i)6-s − 1.75i·7-s + (0.830 − 0.556i)8-s + (−0.993 − 0.109i)9-s + (−0.567 + 0.849i)10-s + (0.160 − 0.160i)11-s + (0.332 + 0.943i)12-s + (−0.183 + 0.183i)13-s + (−0.342 − 1.72i)14-s + (−0.681 − 0.761i)15-s + (0.705 − 0.708i)16-s + 1.18i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $0.942 - 0.333i$ Analytic conductor: $$1.30790$$ Root analytic conductor: $$1.14363$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{48} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :1),\ 0.942 - 0.333i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.64485 + 0.282031i$$ $$L(\frac12)$$ $$\approx$$ $$1.64485 + 0.282031i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.96 + 0.391i)T$$
3 $$1 + (0.164 - 2.99i)T$$
good5 $$1 + (3.61 - 3.61i)T - 25iT^{2}$$
7 $$1 + 12.2iT - 49T^{2}$$
11 $$1 + (-1.76 + 1.76i)T - 121iT^{2}$$
13 $$1 + (2.38 - 2.38i)T - 169iT^{2}$$
17 $$1 - 20.0iT - 289T^{2}$$
19 $$1 + (8.77 - 8.77i)T - 361iT^{2}$$
23 $$1 + 13.1T + 529T^{2}$$
29 $$1 + (6.51 + 6.51i)T + 841iT^{2}$$
31 $$1 - 37.5T + 961T^{2}$$
37 $$1 + (-10.0 - 10.0i)T + 1.36e3iT^{2}$$
41 $$1 + 4.57T + 1.68e3T^{2}$$
43 $$1 + (-21.2 - 21.2i)T + 1.84e3iT^{2}$$
47 $$1 + 54.8iT - 2.20e3T^{2}$$
53 $$1 + (-21.5 + 21.5i)T - 2.80e3iT^{2}$$
59 $$1 + (53.6 - 53.6i)T - 3.48e3iT^{2}$$
61 $$1 + (19.2 - 19.2i)T - 3.72e3iT^{2}$$
67 $$1 + (-31.5 + 31.5i)T - 4.48e3iT^{2}$$
71 $$1 + 65.1T + 5.04e3T^{2}$$
73 $$1 + 50.2iT - 5.32e3T^{2}$$
79 $$1 - 20.9T + 6.24e3T^{2}$$
83 $$1 + (6.35 + 6.35i)T + 6.88e3iT^{2}$$
89 $$1 - 166.T + 7.92e3T^{2}$$
97 $$1 - 139.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$