# Properties

 Label 2-21-21.11-c6-0-13 Degree $2$ Conductor $21$ Sign $-0.348 + 0.937i$ 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 + (13.1 − 7.60i)2-s + (−23.5 − 13.1i)3-s + (83.5 − 144. i)4-s + (−82.1 + 47.4i)5-s + (−410. + 5.43i)6-s + (334. − 73.9i)7-s − 1.56e3i·8-s + (381. + 621. i)9-s + (−721. + 1.24e3i)10-s + (735. + 424. i)11-s + (−3.87e3 + 2.30e3i)12-s + 1.14e3·13-s + (3.84e3 − 3.51e3i)14-s + (2.56e3 − 33.9i)15-s + (−6.56e3 − 1.13e4i)16-s + (−2.38e3 − 1.37e3i)17-s + ⋯
 L(s)  = 1 + (1.64 − 0.950i)2-s + (−0.872 − 0.488i)3-s + (1.30 − 2.26i)4-s + (−0.657 + 0.379i)5-s + (−1.90 + 0.0251i)6-s + (0.976 − 0.215i)7-s − 3.06i·8-s + (0.522 + 0.852i)9-s + (−0.721 + 1.24i)10-s + (0.552 + 0.319i)11-s + (−2.24 + 1.33i)12-s + 0.520·13-s + (1.40 − 1.28i)14-s + (0.758 − 0.0100i)15-s + (−1.60 − 2.77i)16-s + (−0.484 − 0.279i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $-0.348 + 0.937i$ 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: $$(2,\ 21,\ (\ :3),\ -0.348 + 0.937i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$1.58664 - 2.28396i$$ $$L(\frac12)$$ $$\approx$$ $$1.58664 - 2.28396i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (23.5 + 13.1i)T$$
7 $$1 + (-334. + 73.9i)T$$
good2 $$1 + (-13.1 + 7.60i)T + (32 - 55.4i)T^{2}$$
5 $$1 + (82.1 - 47.4i)T + (7.81e3 - 1.35e4i)T^{2}$$
11 $$1 + (-735. - 424. i)T + (8.85e5 + 1.53e6i)T^{2}$$
13 $$1 - 1.14e3T + 4.82e6T^{2}$$
17 $$1 + (2.38e3 + 1.37e3i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (-3.55e3 - 6.15e3i)T + (-2.35e7 + 4.07e7i)T^{2}$$
23 $$1 + (4.81e3 - 2.78e3i)T + (7.40e7 - 1.28e8i)T^{2}$$
29 $$1 - 2.27e4iT - 5.94e8T^{2}$$
31 $$1 + (4.26e3 - 7.37e3i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + (3.59e4 + 6.22e4i)T + (-1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 5.90e4iT - 4.75e9T^{2}$$
43 $$1 + 3.77e4T + 6.32e9T^{2}$$
47 $$1 + (9.76e4 - 5.63e4i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 + (-1.75e5 - 1.01e5i)T + (1.10e10 + 1.91e10i)T^{2}$$
59 $$1 + (-1.50e5 - 8.66e4i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (-2.45e4 - 4.25e4i)T + (-2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (1.56e5 - 2.71e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 - 1.00e5iT - 1.28e11T^{2}$$
73 $$1 + (-1.41e5 + 2.45e5i)T + (-7.56e10 - 1.31e11i)T^{2}$$
79 $$1 + (1.86e5 + 3.22e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 - 2.43e5iT - 3.26e11T^{2}$$
89 $$1 + (-8.90e5 + 5.13e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 + 1.64e6T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$