# Properties

 Label 2-48-3.2-c10-0-8 Degree $2$ Conductor $48$ Sign $0.824 - 0.566i$ Analytic cond. $30.4971$ Root an. cond. $5.52242$ Motivic weight $10$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (200. − 137. i)3-s + 3.63e3i·5-s + 2.32e4·7-s + (2.11e4 − 5.51e4i)9-s + 6.24e4i·11-s − 1.70e5·13-s + (4.99e5 + 7.27e5i)15-s + 2.66e6i·17-s − 7.66e5·19-s + (4.65e6 − 3.19e6i)21-s + 1.40e6i·23-s − 3.41e6·25-s + (−3.34e6 − 1.39e7i)27-s − 4.83e6i·29-s + 4.18e7·31-s + ⋯
 L(s)  = 1 + (0.824 − 0.566i)3-s + 1.16i·5-s + 1.38·7-s + (0.358 − 0.933i)9-s + 0.387i·11-s − 0.458·13-s + (0.657 + 0.957i)15-s + 1.87i·17-s − 0.309·19-s + (1.13 − 0.782i)21-s + 0.218i·23-s − 0.349·25-s + (−0.233 − 0.972i)27-s − 0.235i·29-s + 1.46·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $0.824 - 0.566i$ Analytic conductor: $$30.4971$$ Root analytic conductor: $$5.52242$$ Motivic weight: $$10$$ Rational: no Arithmetic: yes Character: $\chi_{48} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :5),\ 0.824 - 0.566i)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$2.92954 + 0.909572i$$ $$L(\frac12)$$ $$\approx$$ $$2.92954 + 0.909572i$$ $$L(6)$$ 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 + (-200. + 137. i)T$$
good5 $$1 - 3.63e3iT - 9.76e6T^{2}$$
7 $$1 - 2.32e4T + 2.82e8T^{2}$$
11 $$1 - 6.24e4iT - 2.59e10T^{2}$$
13 $$1 + 1.70e5T + 1.37e11T^{2}$$
17 $$1 - 2.66e6iT - 2.01e12T^{2}$$
19 $$1 + 7.66e5T + 6.13e12T^{2}$$
23 $$1 - 1.40e6iT - 4.14e13T^{2}$$
29 $$1 + 4.83e6iT - 4.20e14T^{2}$$
31 $$1 - 4.18e7T + 8.19e14T^{2}$$
37 $$1 - 5.01e7T + 4.80e15T^{2}$$
41 $$1 - 1.49e8iT - 1.34e16T^{2}$$
43 $$1 - 1.98e8T + 2.16e16T^{2}$$
47 $$1 + 1.55e8iT - 5.25e16T^{2}$$
53 $$1 + 4.21e7iT - 1.74e17T^{2}$$
59 $$1 - 2.92e8iT - 5.11e17T^{2}$$
61 $$1 + 5.30e8T + 7.13e17T^{2}$$
67 $$1 + 5.22e8T + 1.82e18T^{2}$$
71 $$1 - 5.71e8iT - 3.25e18T^{2}$$
73 $$1 - 2.18e9T + 4.29e18T^{2}$$
79 $$1 + 1.96e9T + 9.46e18T^{2}$$
83 $$1 + 2.18e9iT - 1.55e19T^{2}$$
89 $$1 - 2.38e8iT - 3.11e19T^{2}$$
97 $$1 + 8.84e9T + 7.37e19T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$