# Properties

 Label 2-21-21.20-c3-0-3 Degree $2$ Conductor $21$ Sign $0.841 + 0.539i$ Analytic cond. $1.23904$ Root an. cond. $1.11312$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.19i·3-s + 8·4-s + (−10 + 15.5i)7-s − 27·9-s − 41.5i·12-s + 62.3i·13-s + 64·16-s − 155. i·19-s + (81 + 51.9i)21-s − 125·25-s + 140. i·27-s + (−80 + 124. i)28-s − 155. i·31-s − 216·36-s − 110·37-s + ⋯
 L(s)  = 1 − 0.999i·3-s + 4-s + (−0.539 + 0.841i)7-s − 9-s − 0.999i·12-s + 1.33i·13-s + 16-s − 1.88i·19-s + (0.841 + 0.539i)21-s − 25-s + 1.00i·27-s + (−0.539 + 0.841i)28-s − 0.903i·31-s − 36-s − 0.488·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $0.841 + 0.539i$ Analytic conductor: $$1.23904$$ Root analytic conductor: $$1.11312$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{21} (20, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :3/2),\ 0.841 + 0.539i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.15884 - 0.339751i$$ $$L(\frac12)$$ $$\approx$$ $$1.15884 - 0.339751i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 5.19iT$$
7 $$1 + (10 - 15.5i)T$$
good2 $$1 - 8T^{2}$$
5 $$1 + 125T^{2}$$
11 $$1 - 1.33e3T^{2}$$
13 $$1 - 62.3iT - 2.19e3T^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 + 155. iT - 6.85e3T^{2}$$
23 $$1 - 1.21e4T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 + 155. iT - 2.97e4T^{2}$$
37 $$1 + 110T + 5.06e4T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 520T + 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 1.48e5T^{2}$$
59 $$1 + 2.05e5T^{2}$$
61 $$1 - 935. iT - 2.26e5T^{2}$$
67 $$1 + 880T + 3.00e5T^{2}$$
71 $$1 - 3.57e5T^{2}$$
73 $$1 + 374. iT - 3.89e5T^{2}$$
79 $$1 - 884T + 4.93e5T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 - 1.37e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$