# Properties

 Label 1-4560-4560.917-r0-0-0 Degree $1$ Conductor $4560$ Sign $0.148 + 0.988i$ Analytic cond. $21.1765$ Root an. cond. $21.1765$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s − 37-s + (−0.939 − 0.342i)41-s + (0.173 + 0.984i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (−0.173 + 0.984i)53-s + (0.642 + 0.766i)59-s + (−0.984 − 0.173i)61-s + ⋯
 L(s)  = 1 + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s − 37-s + (−0.939 − 0.342i)41-s + (0.173 + 0.984i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (−0.173 + 0.984i)53-s + (0.642 + 0.766i)59-s + (−0.984 − 0.173i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$4560$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 19$$ Sign: $0.148 + 0.988i$ Analytic conductor: $$21.1765$$ Root analytic conductor: $$21.1765$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{4560} (917, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 4560,\ (0:\ ),\ 0.148 + 0.988i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.092246732 + 0.9403175781i$$ $$L(\frac12)$$ $$\approx$$ $$1.092246732 + 0.9403175781i$$ $$L(1)$$ $$\approx$$ $$1.050021896 + 0.1519281923i$$ $$L(1)$$ $$\approx$$ $$1.050021896 + 0.1519281923i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
19 $$1$$
good7 $$1 + (0.866 + 0.5i)T$$
11 $$1 + (-0.866 + 0.5i)T$$
13 $$1 + (0.939 - 0.342i)T$$
17 $$1 + (-0.642 - 0.766i)T$$
23 $$1 + (-0.984 - 0.173i)T$$
29 $$1 + (0.642 - 0.766i)T$$
31 $$1 + (-0.5 + 0.866i)T$$
37 $$1 - T$$
41 $$1 + (-0.939 - 0.342i)T$$
43 $$1 + (0.173 + 0.984i)T$$
47 $$1 + (0.642 - 0.766i)T$$
53 $$1 + (-0.173 + 0.984i)T$$
59 $$1 + (0.642 + 0.766i)T$$
61 $$1 + (-0.984 - 0.173i)T$$
67 $$1 + (0.766 + 0.642i)T$$
71 $$1 + (0.173 + 0.984i)T$$
73 $$1 + (0.342 - 0.939i)T$$
79 $$1 + (0.939 + 0.342i)T$$
83 $$1 + (-0.5 + 0.866i)T$$
89 $$1 + (0.939 - 0.342i)T$$
97 $$1 + (0.642 + 0.766i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$