# Properties

 Degree $2$ Conductor $21$ Sign $0.274 - 0.961i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.50i·2-s + (0.822 − 2.88i)3-s − 8.29·4-s − 1.24i·5-s + (10.1 + 2.88i)6-s + 2.64·7-s − 15.0i·8-s + (−7.64 − 4.74i)9-s + 4.35·10-s + 7.01i·11-s + (−6.82 + 23.9i)12-s − 11.6·13-s + 9.27i·14-s + (−3.58 − 1.02i)15-s + 19.5·16-s − 4.52i·17-s + ⋯
 L(s)  = 1 + 1.75i·2-s + (0.274 − 0.961i)3-s − 2.07·4-s − 0.248i·5-s + (1.68 + 0.480i)6-s + 0.377·7-s − 1.88i·8-s + (−0.849 − 0.527i)9-s + 0.435·10-s + 0.637i·11-s + (−0.568 + 1.99i)12-s − 0.895·13-s + 0.662i·14-s + (−0.238 − 0.0681i)15-s + 1.22·16-s − 0.266i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $0.274 - 0.961i$ Motivic weight: $$2$$ Character: $\chi_{21} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :1),\ 0.274 - 0.961i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.689372 + 0.520236i$$ $$L(\frac12)$$ $$\approx$$ $$0.689372 + 0.520236i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.822 + 2.88i)T$$
7 $$1 - 2.64T$$
good2 $$1 - 3.50iT - 4T^{2}$$
5 $$1 + 1.24iT - 25T^{2}$$
11 $$1 - 7.01iT - 121T^{2}$$
13 $$1 + 11.6T + 169T^{2}$$
17 $$1 + 4.52iT - 289T^{2}$$
19 $$1 - 16.2T + 361T^{2}$$
23 $$1 - 25.5iT - 529T^{2}$$
29 $$1 + 9.49iT - 841T^{2}$$
31 $$1 - 28.7T + 961T^{2}$$
37 $$1 + 33.0T + 1.36e3T^{2}$$
41 $$1 + 67.1iT - 1.68e3T^{2}$$
43 $$1 + 24.1T + 1.84e3T^{2}$$
47 $$1 - 33.0iT - 2.20e3T^{2}$$
53 $$1 - 15.1iT - 2.80e3T^{2}$$
59 $$1 + 92.3iT - 3.48e3T^{2}$$
61 $$1 + 57.5T + 3.72e3T^{2}$$
67 $$1 - 15.1T + 4.48e3T^{2}$$
71 $$1 - 70.5iT - 5.04e3T^{2}$$
73 $$1 + 76.7T + 5.32e3T^{2}$$
79 $$1 - 127.T + 6.24e3T^{2}$$
83 $$1 + 74.2iT - 6.88e3T^{2}$$
89 $$1 - 127. iT - 7.92e3T^{2}$$
97 $$1 + 23.1T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$