# Properties

 Label 1-1480-1480.573-r0-0-0 Degree $1$ Conductor $1480$ Sign $0.896 - 0.442i$ 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 + (0.984 + 0.173i)3-s + (0.642 − 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (−0.984 − 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (0.866 + 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (−0.342 − 0.939i)33-s + (0.984 − 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯
 L(s)  = 1 + (0.984 + 0.173i)3-s + (0.642 − 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (−0.984 − 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (0.866 + 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (−0.342 − 0.939i)33-s + (0.984 − 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1480$$    =    $$2^{3} \cdot 5 \cdot 37$$ Sign: $0.896 - 0.442i$ Analytic conductor: $$6.87309$$ Root analytic conductor: $$6.87309$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1480} (573, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1480,\ (0:\ ),\ 0.896 - 0.442i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.560820136 - 0.5970402781i$$ $$L(\frac12)$$ $$\approx$$ $$2.560820136 - 0.5970402781i$$ $$L(1)$$ $$\approx$$ $$1.654403103 - 0.1496897511i$$ $$L(1)$$ $$\approx$$ $$1.654403103 - 0.1496897511i$$

## Euler product

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