# Properties

 Label 2-40e2-5.4-c1-0-5 Degree $2$ Conductor $1600$ Sign $-0.447 - 0.894i$ Analytic cond. $12.7760$ Root an. cond. $3.57436$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·7-s + 3·9-s − 4·11-s + 2i·13-s − 2i·17-s + 4·19-s + 4i·23-s − 2·29-s − 8·31-s + 6i·37-s − 6·41-s + 8i·43-s − 4i·47-s − 9·49-s − 6i·53-s + ⋯
 L(s)  = 1 + 1.51i·7-s + 9-s − 1.20·11-s + 0.554i·13-s − 0.485i·17-s + 0.917·19-s + 0.834i·23-s − 0.371·29-s − 1.43·31-s + 0.986i·37-s − 0.937·41-s + 1.21i·43-s − 0.583i·47-s − 1.28·49-s − 0.824i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$12.7760$$ Root analytic conductor: $$3.57436$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1600} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1600,\ (\ :1/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.276868154$$ $$L(\frac12)$$ $$\approx$$ $$1.276868154$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 - 3T^{2}$$
7 $$1 - 4iT - 7T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 + 2iT - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 + 8T + 31T^{2}$$
37 $$1 - 6iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 - 8iT - 43T^{2}$$
47 $$1 + 4iT - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 6iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 16iT - 83T^{2}$$
89 $$1 - 6T + 89T^{2}$$
97 $$1 - 14iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$