# Properties

 Label 2-1680-5.4-c1-0-21 Degree $2$ Conductor $1680$ Sign $0.447 + 0.894i$ Analytic cond. $13.4148$ Root an. cond. $3.66263$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + (−1 − 2i)5-s + i·7-s − 9-s + 2·11-s + 2i·13-s + (2 − i)15-s − 8i·17-s − 2·19-s − 21-s + (−3 + 4i)25-s − i·27-s + 6·29-s − 6·31-s + 2i·33-s + ⋯
 L(s)  = 1 + 0.577i·3-s + (−0.447 − 0.894i)5-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (0.516 − 0.258i)15-s − 1.94i·17-s − 0.458·19-s − 0.218·21-s + (−0.600 + 0.800i)25-s − 0.192i·27-s + 1.11·29-s − 1.07·31-s + 0.348i·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{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: $$1680$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.447 + 0.894i$ Analytic conductor: $$13.4148$$ Root analytic conductor: $$3.66263$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1680} (1009, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1680,\ (\ :1/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.260578361$$ $$L(\frac12)$$ $$\approx$$ $$1.260578361$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - iT$$
5 $$1 + (1 + 2i)T$$
7 $$1 - iT$$
good11 $$1 - 2T + 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 + 8iT - 17T^{2}$$
19 $$1 + 2T + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 + 6T + 31T^{2}$$
37 $$1 + 8iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 - 4iT - 47T^{2}$$
53 $$1 + 2iT - 53T^{2}$$
59 $$1 + 8T + 59T^{2}$$
61 $$1 - 10T + 61T^{2}$$
67 $$1 + 12iT - 67T^{2}$$
71 $$1 - 14T + 71T^{2}$$
73 $$1 + 10iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 + 16iT - 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$