# Properties

 Label 2-640-40.29-c1-0-0 Degree $2$ Conductor $640$ Sign $-1$ Analytic cond. $5.11042$ Root an. cond. $2.26062$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.16·3-s + 2.23i·5-s + 4.24i·7-s + 7.00·9-s − 7.07i·15-s − 13.4i·21-s + 1.41i·23-s − 5.00·25-s − 12.6·27-s + 8.94i·29-s − 9.48·35-s − 12·41-s − 3.16·43-s + 15.6i·45-s − 9.89i·47-s + ⋯
 L(s)  = 1 − 1.82·3-s + 0.999i·5-s + 1.60i·7-s + 2.33·9-s − 1.82i·15-s − 2.92i·21-s + 0.294i·23-s − 1.00·25-s − 2.43·27-s + 1.66i·29-s − 1.60·35-s − 1.87·41-s − 0.482·43-s + 2.33i·45-s − 1.44i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$640$$    =    $$2^{7} \cdot 5$$ Sign: $-1$ Analytic conductor: $$5.11042$$ Root analytic conductor: $$2.26062$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{640} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 640,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$-0.442641i$$ $$L(\frac12)$$ $$\approx$$ $$-0.442641i$$ $$L(\frac{3}{2})$$ 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 - 2.23iT$$
good3 $$1 + 3.16T + 3T^{2}$$
7 $$1 - 4.24iT - 7T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 - 1.41iT - 23T^{2}$$
29 $$1 - 8.94iT - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 + 12T + 41T^{2}$$
43 $$1 + 3.16T + 43T^{2}$$
47 $$1 + 9.89iT - 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 + 13.4iT - 61T^{2}$$
67 $$1 - 15.8T + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 9.48T + 83T^{2}$$
89 $$1 + 6T + 89T^{2}$$
97 $$1 - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$