# Properties

 Label 2-48-3.2-c18-0-23 Degree $2$ Conductor $48$ Sign $0.215 + 0.976i$ Analytic cond. $98.5853$ Root an. cond. $9.92901$ Motivic weight $18$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.23e3 + 1.92e4i)3-s − 1.14e6i·5-s + 9.81e6·7-s + (−3.51e8 + 1.62e8i)9-s + 2.16e9i·11-s − 1.47e10·13-s + (2.19e10 − 4.83e9i)15-s − 1.36e10i·17-s + 2.38e11·19-s + (4.15e10 + 1.88e11i)21-s − 5.70e11i·23-s + 2.50e12·25-s + (−4.61e12 − 6.06e12i)27-s + 1.35e13i·29-s + 1.17e13·31-s + ⋯
 L(s)  = 1 + (0.215 + 0.976i)3-s − 0.584i·5-s + 0.243·7-s + (−0.907 + 0.420i)9-s + 0.917i·11-s − 1.38·13-s + (0.571 − 0.125i)15-s − 0.115i·17-s + 0.740·19-s + (0.0522 + 0.237i)21-s − 0.316i·23-s + 0.657·25-s + (−0.605 − 0.795i)27-s + 0.932i·29-s + 0.442·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(19-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $0.215 + 0.976i$ Analytic conductor: $$98.5853$$ Root analytic conductor: $$9.92901$$ Motivic weight: $$18$$ Rational: no Arithmetic: yes Character: $\chi_{48} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :9),\ 0.215 + 0.976i)$$

## Particular Values

 $$L(\frac{19}{2})$$ $$\approx$$ $$0.8437045028$$ $$L(\frac12)$$ $$\approx$$ $$0.8437045028$$ $$L(10)$$ 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 + (-4.23e3 - 1.92e4i)T$$
good5 $$1 + 1.14e6iT - 3.81e12T^{2}$$
7 $$1 - 9.81e6T + 1.62e15T^{2}$$
11 $$1 - 2.16e9iT - 5.55e18T^{2}$$
13 $$1 + 1.47e10T + 1.12e20T^{2}$$
17 $$1 + 1.36e10iT - 1.40e22T^{2}$$
19 $$1 - 2.38e11T + 1.04e23T^{2}$$
23 $$1 + 5.70e11iT - 3.24e24T^{2}$$
29 $$1 - 1.35e13iT - 2.10e26T^{2}$$
31 $$1 - 1.17e13T + 6.99e26T^{2}$$
37 $$1 + 1.26e14T + 1.68e28T^{2}$$
41 $$1 + 3.28e14iT - 1.07e29T^{2}$$
43 $$1 + 8.40e14T + 2.52e29T^{2}$$
47 $$1 - 9.97e14iT - 1.25e30T^{2}$$
53 $$1 + 5.45e15iT - 1.08e31T^{2}$$
59 $$1 + 8.03e14iT - 7.50e31T^{2}$$
61 $$1 - 1.04e16T + 1.36e32T^{2}$$
67 $$1 - 1.50e16T + 7.40e32T^{2}$$
71 $$1 + 4.47e16iT - 2.10e33T^{2}$$
73 $$1 + 3.92e16T + 3.46e33T^{2}$$
79 $$1 - 7.20e16T + 1.43e34T^{2}$$
83 $$1 + 1.02e17iT - 3.49e34T^{2}$$
89 $$1 + 5.12e17iT - 1.22e35T^{2}$$
97 $$1 + 7.28e17T + 5.77e35T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$