# Properties

 Label 1-1480-1480.179-r0-0-0 Degree $1$ Conductor $1480$ Sign $0.763 - 0.646i$ Analytic cond. $6.87309$ Root an. cond. $6.87309$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s + 7-s + 9-s − 11-s − i·13-s + i·17-s − i·19-s + 21-s − i·23-s + 27-s − i·29-s − i·31-s − 33-s − i·39-s − 41-s + ⋯
 L(s)  = 1 + 3-s + 7-s + 9-s − 11-s − i·13-s + i·17-s − i·19-s + 21-s − i·23-s + 27-s − i·29-s − i·31-s − 33-s − i·39-s − 41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1480$$    =    $$2^{3} \cdot 5 \cdot 37$$ Sign: $0.763 - 0.646i$ Analytic conductor: $$6.87309$$ Root analytic conductor: $$6.87309$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1480} (179, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1480,\ (0:\ ),\ 0.763 - 0.646i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.333663044 - 0.8555893994i$$ $$L(\frac12)$$ $$\approx$$ $$2.333663044 - 0.8555893994i$$ $$L(1)$$ $$\approx$$ $$1.593407266 - 0.2079021968i$$ $$L(1)$$ $$\approx$$ $$1.593407266 - 0.2079021968i$$

## Euler product

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