# Properties

 Label 2-48-4.3-c8-0-1 Degree $2$ Conductor $48$ Sign $0.866 - 0.5i$ Analytic cond. $19.5541$ Root an. cond. $4.42201$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 46.7i·3-s − 1.08e3·5-s − 426. i·7-s − 2.18e3·9-s + 1.42e4i·11-s + 3.42e4·13-s + 5.07e4i·15-s + 2.00e4·17-s − 1.96e5i·19-s − 1.99e4·21-s + 3.47e5i·23-s + 7.85e5·25-s + 1.02e5i·27-s + 1.00e6·29-s + 1.63e6i·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s − 1.73·5-s − 0.177i·7-s − 0.333·9-s + 0.972i·11-s + 1.19·13-s + 1.00i·15-s + 0.240·17-s − 1.50i·19-s − 0.102·21-s + 1.24i·23-s + 2.01·25-s + 0.192i·27-s + 1.41·29-s + 1.77i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $0.866 - 0.5i$ Analytic conductor: $$19.5541$$ Root analytic conductor: $$4.42201$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{48} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :4),\ 0.866 - 0.5i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.08609 + 0.291017i$$ $$L(\frac12)$$ $$\approx$$ $$1.08609 + 0.291017i$$ $$L(5)$$ 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 + 46.7iT$$
good5 $$1 + 1.08e3T + 3.90e5T^{2}$$
7 $$1 + 426. iT - 5.76e6T^{2}$$
11 $$1 - 1.42e4iT - 2.14e8T^{2}$$
13 $$1 - 3.42e4T + 8.15e8T^{2}$$
17 $$1 - 2.00e4T + 6.97e9T^{2}$$
19 $$1 + 1.96e5iT - 1.69e10T^{2}$$
23 $$1 - 3.47e5iT - 7.83e10T^{2}$$
29 $$1 - 1.00e6T + 5.00e11T^{2}$$
31 $$1 - 1.63e6iT - 8.52e11T^{2}$$
37 $$1 + 7.91e5T + 3.51e12T^{2}$$
41 $$1 - 1.36e6T + 7.98e12T^{2}$$
43 $$1 - 1.50e6iT - 1.16e13T^{2}$$
47 $$1 - 1.49e6iT - 2.38e13T^{2}$$
53 $$1 + 8.94e6T + 6.22e13T^{2}$$
59 $$1 - 8.50e6iT - 1.46e14T^{2}$$
61 $$1 + 1.78e7T + 1.91e14T^{2}$$
67 $$1 + 3.49e7iT - 4.06e14T^{2}$$
71 $$1 - 3.84e7iT - 6.45e14T^{2}$$
73 $$1 - 2.11e7T + 8.06e14T^{2}$$
79 $$1 - 3.67e7iT - 1.51e15T^{2}$$
83 $$1 + 2.94e7iT - 2.25e15T^{2}$$
89 $$1 + 3.70e7T + 3.93e15T^{2}$$
97 $$1 - 1.26e8T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$