# Properties

 Label 2-48-12.11-c15-0-25 Degree $2$ Conductor $48$ Sign $-0.605 + 0.795i$ Analytic cond. $68.4928$ Root an. cond. $8.27604$ Motivic weight $15$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.29e3 − 3.01e3i)3-s − 7.60e4i·5-s − 1.82e6i·7-s + (−3.82e6 − 1.38e7i)9-s + 7.64e7·11-s + 2.54e8·13-s + (−2.29e8 − 1.74e8i)15-s − 7.38e8i·17-s − 5.32e8i·19-s + (−5.51e9 − 4.19e9i)21-s + 1.70e10·23-s + 2.47e10·25-s + (−5.04e10 − 2.01e10i)27-s − 1.64e11i·29-s + 1.19e11i·31-s + ⋯
 L(s)  = 1 + (0.605 − 0.795i)3-s − 0.435i·5-s − 0.839i·7-s + (−0.266 − 0.963i)9-s + 1.18·11-s + 1.12·13-s + (−0.346 − 0.263i)15-s − 0.436i·17-s − 0.136i·19-s + (−0.668 − 0.508i)21-s + 1.04·23-s + 0.810·25-s + (−0.928 − 0.371i)27-s − 1.76i·29-s + 0.780i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(16-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $-0.605 + 0.795i$ Analytic conductor: $$68.4928$$ Root analytic conductor: $$8.27604$$ Motivic weight: $$15$$ Rational: no Arithmetic: yes Character: $\chi_{48} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :15/2),\ -0.605 + 0.795i)$$

## Particular Values

 $$L(8)$$ $$\approx$$ $$3.043649284$$ $$L(\frac12)$$ $$\approx$$ $$3.043649284$$ $$L(\frac{17}{2})$$ 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 + (-2.29e3 + 3.01e3i)T$$
good5 $$1 + 7.60e4iT - 3.05e10T^{2}$$
7 $$1 + 1.82e6iT - 4.74e12T^{2}$$
11 $$1 - 7.64e7T + 4.17e15T^{2}$$
13 $$1 - 2.54e8T + 5.11e16T^{2}$$
17 $$1 + 7.38e8iT - 2.86e18T^{2}$$
19 $$1 + 5.32e8iT - 1.51e19T^{2}$$
23 $$1 - 1.70e10T + 2.66e20T^{2}$$
29 $$1 + 1.64e11iT - 8.62e21T^{2}$$
31 $$1 - 1.19e11iT - 2.34e22T^{2}$$
37 $$1 + 1.78e11T + 3.33e23T^{2}$$
41 $$1 - 5.29e11iT - 1.55e24T^{2}$$
43 $$1 - 5.71e11iT - 3.17e24T^{2}$$
47 $$1 + 1.37e12T + 1.20e25T^{2}$$
53 $$1 - 1.30e13iT - 7.31e25T^{2}$$
59 $$1 + 6.31e11T + 3.65e26T^{2}$$
61 $$1 + 1.44e13T + 6.02e26T^{2}$$
67 $$1 + 8.74e13iT - 2.46e27T^{2}$$
71 $$1 + 8.26e13T + 5.87e27T^{2}$$
73 $$1 - 6.67e13T + 8.90e27T^{2}$$
79 $$1 + 1.93e14iT - 2.91e28T^{2}$$
83 $$1 + 4.13e14T + 6.11e28T^{2}$$
89 $$1 - 5.61e14iT - 1.74e29T^{2}$$
97 $$1 + 3.75e14T + 6.33e29T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$