# Properties

 Label 2-21-21.20-c29-0-4 Degree $2$ Conductor $21$ Sign $0.793 - 0.609i$ Analytic cond. $111.883$ Root an. cond. $10.5775$ Motivic weight $29$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4.26e4i·2-s + (−4.19e6 + 7.14e6i)3-s − 1.28e9·4-s − 1.62e10·5-s + (3.04e11 + 1.78e11i)6-s + (1.66e12 + 6.74e11i)7-s + 3.16e13i·8-s + (−3.35e13 − 5.98e13i)9-s + 6.92e14i·10-s + 1.89e15i·11-s + (5.36e15 − 9.14e15i)12-s − 1.67e16i·13-s + (2.87e16 − 7.08e16i)14-s + (6.80e16 − 1.16e17i)15-s + 6.63e17·16-s + 1.70e17·17-s + ⋯
 L(s)  = 1 − 1.83i·2-s + (−0.505 + 0.862i)3-s − 2.38·4-s − 1.18·5-s + (1.58 + 0.930i)6-s + (0.926 + 0.376i)7-s + 2.54i·8-s + (−0.488 − 0.872i)9-s + 2.18i·10-s + 1.50i·11-s + (1.20 − 2.05i)12-s − 1.17i·13-s + (0.691 − 1.70i)14-s + (0.601 − 1.02i)15-s + 2.30·16-s + 0.246·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(30-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $0.793 - 0.609i$ Analytic conductor: $$111.883$$ Root analytic conductor: $$10.5775$$ Motivic weight: $$29$$ Rational: no Arithmetic: yes Character: $\chi_{21} (20, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :29/2),\ 0.793 - 0.609i)$$

## Particular Values

 $$L(15)$$ $$\approx$$ $$0.3558104268$$ $$L(\frac12)$$ $$\approx$$ $$0.3558104268$$ $$L(\frac{31}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (4.19e6 - 7.14e6i)T$$
7 $$1 + (-1.66e12 - 6.74e11i)T$$
good2 $$1 + 4.26e4iT - 5.36e8T^{2}$$
5 $$1 + 1.62e10T + 1.86e20T^{2}$$
11 $$1 - 1.89e15iT - 1.58e30T^{2}$$
13 $$1 + 1.67e16iT - 2.01e32T^{2}$$
17 $$1 - 1.70e17T + 4.81e35T^{2}$$
19 $$1 + 2.87e18iT - 1.21e37T^{2}$$
23 $$1 + 8.58e18iT - 3.09e39T^{2}$$
29 $$1 - 1.33e21iT - 2.56e42T^{2}$$
31 $$1 + 8.21e21iT - 1.77e43T^{2}$$
37 $$1 + 6.15e22T + 3.00e45T^{2}$$
41 $$1 + 2.49e21T + 5.89e46T^{2}$$
43 $$1 - 8.52e23T + 2.34e47T^{2}$$
47 $$1 + 2.18e24T + 3.09e48T^{2}$$
53 $$1 + 1.34e25iT - 1.00e50T^{2}$$
59 $$1 + 2.12e25T + 2.26e51T^{2}$$
61 $$1 - 5.23e25iT - 5.95e51T^{2}$$
67 $$1 - 1.73e26T + 9.04e52T^{2}$$
71 $$1 + 1.36e26iT - 4.85e53T^{2}$$
73 $$1 + 1.03e27iT - 1.08e54T^{2}$$
79 $$1 - 1.92e27T + 1.07e55T^{2}$$
83 $$1 - 5.55e27T + 4.50e55T^{2}$$
89 $$1 + 3.90e26T + 3.40e56T^{2}$$
97 $$1 + 2.98e28iT - 4.13e57T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$