# Properties

 Label 2-48-12.11-c3-0-3 Degree $2$ Conductor $48$ Sign $-0.182 + 0.983i$ Analytic cond. $2.83209$ Root an. cond. $1.68288$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.89 + 1.73i)3-s − 16.9i·5-s − 17.3i·7-s + (20.9 − 16.9i)9-s − 29.3·11-s − 26·13-s + (29.3 + 83.1i)15-s + 67.8i·17-s − 107. i·19-s + (30 + 84.8i)21-s + 176.·23-s − 162.·25-s + (−73.4 + 119. i)27-s + 16.9i·29-s − 31.1i·31-s + ⋯
 L(s)  = 1 + (−0.942 + 0.333i)3-s − 1.51i·5-s − 0.935i·7-s + (0.777 − 0.628i)9-s − 0.805·11-s − 0.554·13-s + (0.505 + 1.43i)15-s + 0.968i·17-s − 1.29i·19-s + (0.311 + 0.881i)21-s + 1.59·23-s − 1.30·25-s + (−0.523 + 0.851i)27-s + 0.108i·29-s − 0.180i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $-0.182 + 0.983i$ Analytic conductor: $$2.83209$$ Root analytic conductor: $$1.68288$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{48} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :3/2),\ -0.182 + 0.983i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.519043 - 0.624400i$$ $$L(\frac12)$$ $$\approx$$ $$0.519043 - 0.624400i$$ $$L(\frac{5}{2})$$ 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 + (4.89 - 1.73i)T$$
good5 $$1 + 16.9iT - 125T^{2}$$
7 $$1 + 17.3iT - 343T^{2}$$
11 $$1 + 29.3T + 1.33e3T^{2}$$
13 $$1 + 26T + 2.19e3T^{2}$$
17 $$1 - 67.8iT - 4.91e3T^{2}$$
19 $$1 + 107. iT - 6.85e3T^{2}$$
23 $$1 - 176.T + 1.21e4T^{2}$$
29 $$1 - 16.9iT - 2.43e4T^{2}$$
31 $$1 + 31.1iT - 2.97e4T^{2}$$
37 $$1 - 206T + 5.06e4T^{2}$$
41 $$1 + 305. iT - 6.89e4T^{2}$$
43 $$1 + 93.5iT - 7.95e4T^{2}$$
47 $$1 + 117.T + 1.03e5T^{2}$$
53 $$1 - 50.9iT - 1.48e5T^{2}$$
59 $$1 - 558.T + 2.05e5T^{2}$$
61 $$1 - 278T + 2.26e5T^{2}$$
67 $$1 - 890. iT - 3.00e5T^{2}$$
71 $$1 - 58.7T + 3.57e5T^{2}$$
73 $$1 + 422T + 3.89e5T^{2}$$
79 $$1 + 668. iT - 4.93e5T^{2}$$
83 $$1 - 29.3T + 5.71e5T^{2}$$
89 $$1 + 373. iT - 7.04e5T^{2}$$
97 $$1 + 1.07e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$