# Properties

 Label 2-21-7.3-c6-0-3 Degree $2$ Conductor $21$ Sign $-0.659 - 0.751i$ Analytic cond. $4.83113$ Root an. cond. $2.19798$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (6.58 + 11.4i)2-s + (13.5 + 7.79i)3-s + (−54.8 + 94.9i)4-s + (−68.9 + 39.7i)5-s + 205. i·6-s + (284. − 191. i)7-s − 601.·8-s + (121.5 + 210. i)9-s + (−908. − 524. i)10-s + (−411. + 712. i)11-s + (−1.48e3 + 854. i)12-s − 2.42e3i·13-s + (4.06e3 + 1.97e3i)14-s − 1.24e3·15-s + (−456. − 791. i)16-s + (6.75e3 + 3.89e3i)17-s + ⋯
 L(s)  = 1 + (0.823 + 1.42i)2-s + (0.5 + 0.288i)3-s + (−0.856 + 1.48i)4-s + (−0.551 + 0.318i)5-s + 0.951i·6-s + (0.828 − 0.559i)7-s − 1.17·8-s + (0.166 + 0.288i)9-s + (−0.908 − 0.524i)10-s + (−0.308 + 0.534i)11-s + (−0.856 + 0.494i)12-s − 1.10i·13-s + (1.48 + 0.721i)14-s − 0.367·15-s + (−0.111 − 0.193i)16-s + (1.37 + 0.793i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $-0.659 - 0.751i$ Analytic conductor: $$4.83113$$ Root analytic conductor: $$2.19798$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{21} (10, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :3),\ -0.659 - 0.751i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$1.01761 + 2.24836i$$ $$L(\frac12)$$ $$\approx$$ $$1.01761 + 2.24836i$$ $$L(4)$$ 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 + (-284. + 191. i)T$$
good2 $$1 + (-6.58 - 11.4i)T + (-32 + 55.4i)T^{2}$$
5 $$1 + (68.9 - 39.7i)T + (7.81e3 - 1.35e4i)T^{2}$$
11 $$1 + (411. - 712. i)T + (-8.85e5 - 1.53e6i)T^{2}$$
13 $$1 + 2.42e3iT - 4.82e6T^{2}$$
17 $$1 + (-6.75e3 - 3.89e3i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (-5.78e3 + 3.34e3i)T + (2.35e7 - 4.07e7i)T^{2}$$
23 $$1 + (9.41e3 + 1.63e4i)T + (-7.40e7 + 1.28e8i)T^{2}$$
29 $$1 - 1.38e4T + 5.94e8T^{2}$$
31 $$1 + (2.41e4 + 1.39e4i)T + (4.43e8 + 7.68e8i)T^{2}$$
37 $$1 + (3.98e4 + 6.90e4i)T + (-1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 5.91e4iT - 4.75e9T^{2}$$
43 $$1 + 9.18e4T + 6.32e9T^{2}$$
47 $$1 + (4.34e3 - 2.50e3i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 + (9.31e4 - 1.61e5i)T + (-1.10e10 - 1.91e10i)T^{2}$$
59 $$1 + (-1.95e5 - 1.12e5i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (1.25e5 - 7.21e4i)T + (2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (-1.17e5 + 2.03e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 - 9.62e4T + 1.28e11T^{2}$$
73 $$1 + (-2.38e5 - 1.37e5i)T + (7.56e10 + 1.31e11i)T^{2}$$
79 $$1 + (-3.40e5 - 5.90e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 - 1.28e5iT - 3.26e11T^{2}$$
89 $$1 + (3.22e5 - 1.86e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 + 6.20e5iT - 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$