# Properties

 Label 2-21-21.17-c5-0-4 Degree $2$ Conductor $21$ Sign $0.473 - 0.880i$ Analytic cond. $3.36806$ Root an. cond. $1.83522$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (13.5 + 7.79i)3-s + (−16 + 27.7i)4-s + (105.5 + 75.3i)7-s + (121.5 + 210. i)9-s + (−432 + 249. i)12-s − 1.14e3i·13-s + (−511. − 886. i)16-s + (−139.5 + 80.5i)19-s + (837 + 1.83e3i)21-s + (1.56e3 − 2.70e3i)25-s + 3.78e3i·27-s + (−3.77e3 + 1.71e3i)28-s + (8.96e3 + 5.17e3i)31-s − 7.77e3·36-s + (−3.33e3 − 5.76e3i)37-s + ⋯
 L(s)  = 1 + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.813 + 0.581i)7-s + (0.5 + 0.866i)9-s + (−0.866 + 0.499i)12-s − 1.87i·13-s + (−0.499 − 0.866i)16-s + (−0.0886 + 0.0511i)19-s + (0.414 + 0.910i)21-s + (0.5 − 0.866i)25-s + 1.00i·27-s + (−0.910 + 0.414i)28-s + (1.67 + 0.967i)31-s − 36-s + (−0.399 − 0.692i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $0.473 - 0.880i$ Analytic conductor: $$3.36806$$ Root analytic conductor: $$1.83522$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{21} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :5/2),\ 0.473 - 0.880i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.46231 + 0.873953i$$ $$L(\frac12)$$ $$\approx$$ $$1.46231 + 0.873953i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-13.5 - 7.79i)T$$
7 $$1 + (-105.5 - 75.3i)T$$
good2 $$1 + (16 - 27.7i)T^{2}$$
5 $$1 + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + 1.14e3iT - 3.71e5T^{2}$$
17 $$1 + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (139.5 - 80.5i)T + (1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (3.21e6 - 5.57e6i)T^{2}$$
29 $$1 - 2.05e7T^{2}$$
31 $$1 + (-8.96e3 - 5.17e3i)T + (1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (3.33e3 + 5.76e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 1.15e8T^{2}$$
43 $$1 + 2.24e4T + 1.47e8T^{2}$$
47 $$1 + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (3.76e4 - 2.17e4i)T + (4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-1.89e4 + 3.28e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 - 1.80e9T^{2}$$
73 $$1 + (4.05e4 + 2.33e4i)T + (1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (4.54e4 + 7.86e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 3.93e9T^{2}$$
89 $$1 + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 1.27e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$