# Properties

 Label 2-1584-44.43-c1-0-25 Degree $2$ Conductor $1584$ Sign $-0.296 + 0.955i$ Analytic cond. $12.6483$ Root an. cond. $3.55644$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.732·5-s − 2.36·7-s + (3.23 − 0.732i)11-s − 2.36i·13-s − 6.46i·17-s − 6.46·19-s + 4.73i·23-s − 4.46·25-s − 2i·31-s − 1.73·35-s + 7.46·37-s − 4.73i·41-s − 6.46·43-s − 6.19i·47-s − 1.39·49-s + ⋯
 L(s)  = 1 + 0.327·5-s − 0.895·7-s + (0.975 − 0.220i)11-s − 0.656i·13-s − 1.56i·17-s − 1.48·19-s + 0.986i·23-s − 0.892·25-s − 0.359i·31-s − 0.293·35-s + 1.22·37-s − 0.739i·41-s − 0.986·43-s − 0.903i·47-s − 0.198·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1584$$    =    $$2^{4} \cdot 3^{2} \cdot 11$$ Sign: $-0.296 + 0.955i$ Analytic conductor: $$12.6483$$ Root analytic conductor: $$3.55644$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1584} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1584,\ (\ :1/2),\ -0.296 + 0.955i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.087277979$$ $$L(\frac12)$$ $$\approx$$ $$1.087277979$$ $$L(\frac{3}{2})$$ 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$$
11 $$1 + (-3.23 + 0.732i)T$$
good5 $$1 - 0.732T + 5T^{2}$$
7 $$1 + 2.36T + 7T^{2}$$
13 $$1 + 2.36iT - 13T^{2}$$
17 $$1 + 6.46iT - 17T^{2}$$
19 $$1 + 6.46T + 19T^{2}$$
23 $$1 - 4.73iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 2iT - 31T^{2}$$
37 $$1 - 7.46T + 37T^{2}$$
41 $$1 + 4.73iT - 41T^{2}$$
43 $$1 + 6.46T + 43T^{2}$$
47 $$1 + 6.19iT - 47T^{2}$$
53 $$1 - 7.26T + 53T^{2}$$
59 $$1 - 2.53iT - 59T^{2}$$
61 $$1 + 15.3iT - 61T^{2}$$
67 $$1 + 10.3iT - 67T^{2}$$
71 $$1 + 7.26iT - 71T^{2}$$
73 $$1 + 4.73iT - 73T^{2}$$
79 $$1 - 2.36T + 79T^{2}$$
83 $$1 + 4.73T + 83T^{2}$$
89 $$1 + 10.3T + 89T^{2}$$
97 $$1 + 1.46T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$