# Properties

 Label 2-2888-152.53-c0-0-1 Degree $2$ Conductor $2888$ Sign $0.378 - 0.925i$ Analytic cond. $1.44129$ Root an. cond. $1.20054$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.5 − 0.866i)7-s + (0.500 + 0.866i)8-s + (−0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.766 + 0.642i)24-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯
 L(s)  = 1 + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.5 − 0.866i)7-s + (0.500 + 0.866i)8-s + (−0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.766 + 0.642i)24-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2888$$    =    $$2^{3} \cdot 19^{2}$$ Sign: $0.378 - 0.925i$ Analytic conductor: $$1.44129$$ Root analytic conductor: $$1.20054$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2888} (2789, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2888,\ (\ :0),\ 0.378 - 0.925i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.592226248$$ $$L(\frac12)$$ $$\approx$$ $$2.592226248$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.939 - 0.342i)T$$
19 $$1$$
good3 $$1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}$$
5 $$1 + (-0.173 + 0.984i)T^{2}$$
7 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
11 $$1 + (0.5 - 0.866i)T^{2}$$
13 $$1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}$$
17 $$1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}$$
23 $$1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}$$
29 $$1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}$$
31 $$1 + (0.5 + 0.866i)T^{2}$$
37 $$1 + 2T + T^{2}$$
41 $$1 + (0.939 - 0.342i)T^{2}$$
43 $$1 + (-0.173 + 0.984i)T^{2}$$
47 $$1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2}$$
53 $$1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}$$
59 $$1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2}$$
61 $$1 + (-0.173 - 0.984i)T^{2}$$
67 $$1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}$$
71 $$1 + (-0.173 + 0.984i)T^{2}$$
73 $$1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}$$
79 $$1 + (0.939 - 0.342i)T^{2}$$
83 $$1 + (0.5 + 0.866i)T^{2}$$
89 $$1 + (0.939 + 0.342i)T^{2}$$
97 $$1 + (-0.766 - 0.642i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$