# Properties

 Label 2-420-60.59-c1-0-47 Degree $2$ Conductor $420$ Sign $0.367 + 0.930i$ Analytic cond. $3.35371$ Root an. cond. $1.83131$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.621 − 1.27i)2-s + (1.72 + 0.121i)3-s + (−1.22 + 1.57i)4-s + (1 − 2i)5-s + (−0.919 − 2.27i)6-s + 7-s + (2.76 + 0.578i)8-s + (2.97 + 0.419i)9-s + (−3.16 − 0.0276i)10-s + 3.45·11-s + (−2.31 + 2.57i)12-s + 4.83i·13-s + (−0.621 − 1.27i)14-s + (1.97 − 3.33i)15-s + (−0.985 − 3.87i)16-s − 5.94·17-s + ⋯
 L(s)  = 1 + (−0.439 − 0.898i)2-s + (0.997 + 0.0700i)3-s + (−0.613 + 0.789i)4-s + (0.447 − 0.894i)5-s + (−0.375 − 0.926i)6-s + 0.377·7-s + (0.978 + 0.204i)8-s + (0.990 + 0.139i)9-s + (−0.999 − 0.00874i)10-s + 1.04·11-s + (−0.667 + 0.744i)12-s + 1.34i·13-s + (−0.166 − 0.339i)14-s + (0.508 − 0.860i)15-s + (−0.246 − 0.969i)16-s − 1.44·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$420$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.367 + 0.930i$ Analytic conductor: $$3.35371$$ Root analytic conductor: $$1.83131$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{420} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 420,\ (\ :1/2),\ 0.367 + 0.930i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.39004 - 0.945651i$$ $$L(\frac12)$$ $$\approx$$ $$1.39004 - 0.945651i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.621 + 1.27i)T$$
3 $$1 + (-1.72 - 0.121i)T$$
5 $$1 + (-1 + 2i)T$$
7 $$1 - T$$
good11 $$1 - 3.45T + 11T^{2}$$
13 $$1 - 4.83iT - 13T^{2}$$
17 $$1 + 5.94T + 17T^{2}$$
19 $$1 + 1.08iT - 19T^{2}$$
23 $$1 - 0.596iT - 23T^{2}$$
29 $$1 + 4.83iT - 29T^{2}$$
31 $$1 + 9.56iT - 31T^{2}$$
37 $$1 - 2.91iT - 37T^{2}$$
41 $$1 - 6.91iT - 41T^{2}$$
43 $$1 - 7.39T + 43T^{2}$$
47 $$1 + 0.242iT - 47T^{2}$$
53 $$1 + 11.8T + 53T^{2}$$
59 $$1 - 3.25T + 59T^{2}$$
61 $$1 + 12.6T + 61T^{2}$$
67 $$1 + 5.71T + 67T^{2}$$
71 $$1 + 8T + 71T^{2}$$
73 $$1 - 8iT - 73T^{2}$$
79 $$1 - 0.949iT - 79T^{2}$$
83 $$1 - 16.9iT - 83T^{2}$$
89 $$1 - 5.23iT - 89T^{2}$$
97 $$1 + 3.30iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$