# Properties

 Label 2-21-21.11-c32-0-30 Degree $2$ Conductor $21$ Sign $-0.706 - 0.707i$ Analytic cond. $136.219$ Root an. cond. $11.6713$ Motivic weight $32$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.15e7 + 3.72e7i)3-s + (−2.14e9 + 3.71e9i)4-s + (3.25e13 + 6.57e12i)7-s + (−9.26e14 − 1.60e15i)9-s + (−9.24e16 − 1.60e17i)12-s + 1.22e17·13-s + (−9.22e18 − 1.59e19i)16-s + (3.91e18 + 6.77e18i)19-s + (−9.46e20 + 1.07e21i)21-s + (−1.16e22 + 2.01e22i)25-s + 7.97e22·27-s + (−9.44e22 + 1.07e23i)28-s + (7.06e23 − 1.22e24i)31-s + (7.95e24 + 5.36e8i)36-s + (1.07e25 + 1.86e25i)37-s + ⋯
 L(s)  = 1 + (−0.499 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.980 + 0.197i)7-s + (−0.5 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 0.183·13-s + (−0.499 − 0.866i)16-s + (0.0135 + 0.0234i)19-s + (−0.661 + 0.749i)21-s + (−0.5 + 0.866i)25-s + 27-s + (−0.661 + 0.749i)28-s + (0.971 − 1.68i)31-s + 36-s + (0.873 + 1.51i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(33-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $-0.706 - 0.707i$ Analytic conductor: $$136.219$$ Root analytic conductor: $$11.6713$$ Motivic weight: $$32$$ Rational: no Arithmetic: yes Character: $\chi_{21} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :16),\ -0.706 - 0.707i)$$

## Particular Values

 $$L(\frac{33}{2})$$ $$\approx$$ $$1.621370663$$ $$L(\frac12)$$ $$\approx$$ $$1.621370663$$ $$L(17)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.15e7 - 3.72e7i)T$$
7 $$1 + (-3.25e13 - 6.57e12i)T$$
good2 $$1 + (2.14e9 - 3.71e9i)T^{2}$$
5 $$1 + (1.16e22 - 2.01e22i)T^{2}$$
11 $$1 + (1.05e33 + 1.82e33i)T^{2}$$
13 $$1 - 1.22e17T + 4.42e35T^{2}$$
17 $$1 + (1.18e39 + 2.05e39i)T^{2}$$
19 $$1 + (-3.91e18 - 6.77e18i)T + (-4.15e40 + 7.20e40i)T^{2}$$
23 $$1 + (1.88e43 - 3.25e43i)T^{2}$$
29 $$1 - 6.26e46T^{2}$$
31 $$1 + (-7.06e23 + 1.22e24i)T + (-2.64e47 - 4.58e47i)T^{2}$$
37 $$1 + (-1.07e25 - 1.86e25i)T + (-7.61e49 + 1.31e50i)T^{2}$$
41 $$1 - 4.06e51T^{2}$$
43 $$1 - 2.72e26T + 1.86e52T^{2}$$
47 $$1 + (1.60e53 - 2.78e53i)T^{2}$$
53 $$1 + (7.51e54 + 1.30e55i)T^{2}$$
59 $$1 + (2.32e56 + 4.02e56i)T^{2}$$
61 $$1 + (-1.97e28 - 3.42e28i)T + (-6.75e56 + 1.16e57i)T^{2}$$
67 $$1 + (2.65e27 - 4.60e27i)T + (-1.35e58 - 2.35e58i)T^{2}$$
71 $$1 - 1.73e59T^{2}$$
73 $$1 + (3.83e29 - 6.64e29i)T + (-2.11e59 - 3.66e59i)T^{2}$$
79 $$1 + (1.01e30 + 1.75e30i)T + (-2.64e60 + 4.58e60i)T^{2}$$
83 $$1 - 2.57e61T^{2}$$
89 $$1 + (1.20e62 - 2.07e62i)T^{2}$$
97 $$1 - 1.21e32T + 3.77e63T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$