# Properties

 Label 2-21-21.11-c6-0-2 Degree $2$ Conductor $21$ Sign $-0.591 - 0.806i$ 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 + (5.69 − 3.28i)2-s + (−24.6 + 11.0i)3-s + (−10.3 + 17.9i)4-s + (−57.4 + 33.1i)5-s + (−104. + 143. i)6-s + (−329. + 93.7i)7-s + 557. i·8-s + (486. − 542. i)9-s + (−218. + 377. i)10-s + (603. + 348. i)11-s + (57.8 − 556. i)12-s − 824.·13-s + (−1.57e3 + 1.61e3i)14-s + (1.05e3 − 1.45e3i)15-s + (1.16e3 + 2.02e3i)16-s + (−4.78e3 − 2.76e3i)17-s + ⋯
 L(s)  = 1 + (0.712 − 0.411i)2-s + (−0.913 + 0.407i)3-s + (−0.161 + 0.280i)4-s + (−0.459 + 0.265i)5-s + (−0.482 + 0.665i)6-s + (−0.961 + 0.273i)7-s + 1.08i·8-s + (0.667 − 0.744i)9-s + (−0.218 + 0.377i)10-s + (0.453 + 0.261i)11-s + (0.0334 − 0.322i)12-s − 0.375·13-s + (−0.572 + 0.590i)14-s + (0.311 − 0.429i)15-s + (0.285 + 0.494i)16-s + (−0.973 − 0.562i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $-0.591 - 0.806i$ Analytic conductor: $$4.83113$$ Root analytic conductor: $$2.19798$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{21} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :3),\ -0.591 - 0.806i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.360283 + 0.711536i$$ $$L(\frac12)$$ $$\approx$$ $$0.360283 + 0.711536i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (24.6 - 11.0i)T$$
7 $$1 + (329. - 93.7i)T$$
good2 $$1 + (-5.69 + 3.28i)T + (32 - 55.4i)T^{2}$$
5 $$1 + (57.4 - 33.1i)T + (7.81e3 - 1.35e4i)T^{2}$$
11 $$1 + (-603. - 348. i)T + (8.85e5 + 1.53e6i)T^{2}$$
13 $$1 + 824.T + 4.82e6T^{2}$$
17 $$1 + (4.78e3 + 2.76e3i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (-2.52e3 - 4.37e3i)T + (-2.35e7 + 4.07e7i)T^{2}$$
23 $$1 + (-1.72e4 + 9.94e3i)T + (7.40e7 - 1.28e8i)T^{2}$$
29 $$1 - 2.34e4iT - 5.94e8T^{2}$$
31 $$1 + (2.31e4 - 4.01e4i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + (-2.41e4 - 4.18e4i)T + (-1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 1.59e4iT - 4.75e9T^{2}$$
43 $$1 - 2.17e4T + 6.32e9T^{2}$$
47 $$1 + (3.30e4 - 1.90e4i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 + (1.93e5 + 1.11e5i)T + (1.10e10 + 1.91e10i)T^{2}$$
59 $$1 + (-2.71e5 - 1.56e5i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (1.26e5 + 2.19e5i)T + (-2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (-8.29e4 + 1.43e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + 2.98e5iT - 1.28e11T^{2}$$
73 $$1 + (2.21e3 - 3.83e3i)T + (-7.56e10 - 1.31e11i)T^{2}$$
79 $$1 + (-3.10e5 - 5.37e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + 4.14e5iT - 3.26e11T^{2}$$
89 $$1 + (6.49e5 - 3.75e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 - 2.12e5T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$