# Properties

 Label 2-21-7.3-c6-0-2 Degree $2$ Conductor $21$ Sign $-0.984 - 0.178i$ 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 + (5.80 + 10.0i)2-s + (−13.5 − 7.79i)3-s + (−35.4 + 61.4i)4-s + (−165. + 95.4i)5-s − 181. i·6-s + (−103. + 327. i)7-s − 80.4·8-s + (121.5 + 210. i)9-s + (−1.92e3 − 1.10e3i)10-s + (1.02e3 − 1.77e3i)11-s + (957. − 552. i)12-s + 3.05e3i·13-s + (−3.89e3 + 861. i)14-s + 2.97e3·15-s + (1.80e3 + 3.12e3i)16-s + (2.46e3 + 1.42e3i)17-s + ⋯
 L(s)  = 1 + (0.725 + 1.25i)2-s + (−0.5 − 0.288i)3-s + (−0.554 + 0.959i)4-s + (−1.32 + 0.763i)5-s − 0.838i·6-s + (−0.301 + 0.953i)7-s − 0.157·8-s + (0.166 + 0.288i)9-s + (−1.92 − 1.10i)10-s + (0.772 − 1.33i)11-s + (0.554 − 0.319i)12-s + 1.39i·13-s + (−1.41 + 0.313i)14-s + 0.881·15-s + (0.439 + 0.762i)16-s + (0.502 + 0.290i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.178i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.984 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $-0.984 - 0.178i$ 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: $$(2,\ 21,\ (\ :3),\ -0.984 - 0.178i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.117285 + 1.30676i$$ $$L(\frac12)$$ $$\approx$$ $$0.117285 + 1.30676i$$ $$L(4)$$ 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 + (103. - 327. i)T$$
good2 $$1 + (-5.80 - 10.0i)T + (-32 + 55.4i)T^{2}$$
5 $$1 + (165. - 95.4i)T + (7.81e3 - 1.35e4i)T^{2}$$
11 $$1 + (-1.02e3 + 1.77e3i)T + (-8.85e5 - 1.53e6i)T^{2}$$
13 $$1 - 3.05e3iT - 4.82e6T^{2}$$
17 $$1 + (-2.46e3 - 1.42e3i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (3.42e3 - 1.97e3i)T + (2.35e7 - 4.07e7i)T^{2}$$
23 $$1 + (330. + 572. i)T + (-7.40e7 + 1.28e8i)T^{2}$$
29 $$1 + 9.28e3T + 5.94e8T^{2}$$
31 $$1 + (-2.42e3 - 1.40e3i)T + (4.43e8 + 7.68e8i)T^{2}$$
37 $$1 + (-1.84e4 - 3.19e4i)T + (-1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 6.79e4iT - 4.75e9T^{2}$$
43 $$1 - 1.23e4T + 6.32e9T^{2}$$
47 $$1 + (-1.27e5 + 7.38e4i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 + (1.09e5 - 1.90e5i)T + (-1.10e10 - 1.91e10i)T^{2}$$
59 $$1 + (-1.66e5 - 9.62e4i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (-2.88e5 + 1.66e5i)T + (2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (1.74e5 - 3.02e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 - 3.05e5T + 1.28e11T^{2}$$
73 $$1 + (2.04e5 + 1.18e5i)T + (7.56e10 + 1.31e11i)T^{2}$$
79 $$1 + (-2.93e5 - 5.07e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 - 1.06e5iT - 3.26e11T^{2}$$
89 $$1 + (-1.13e4 + 6.52e3i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 - 2.05e5iT - 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$