# Properties

 Label 2-21e2-63.25-c1-0-34 Degree $2$ Conductor $441$ Sign $-0.888 + 0.458i$ Analytic cond. $3.52140$ Root an. cond. $1.87654$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − 1.73i·3-s − 4-s + (−0.5 + 0.866i)5-s − 1.73i·6-s − 3·8-s − 2.99·9-s + (−0.5 + 0.866i)10-s + (−2.5 − 4.33i)11-s + 1.73i·12-s + (−2.5 − 4.33i)13-s + (1.49 + 0.866i)15-s − 16-s + (1.5 − 2.59i)17-s − 2.99·18-s + (0.5 + 0.866i)19-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.999i·3-s − 0.5·4-s + (−0.223 + 0.387i)5-s − 0.707i·6-s − 1.06·8-s − 0.999·9-s + (−0.158 + 0.273i)10-s + (−0.753 − 1.30i)11-s + 0.499i·12-s + (−0.693 − 1.20i)13-s + (0.387 + 0.223i)15-s − 0.250·16-s + (0.363 − 0.630i)17-s − 0.707·18-s + (0.114 + 0.198i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-0.888 + 0.458i$ Analytic conductor: $$3.52140$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{441} (214, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ -0.888 + 0.458i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.212143 - 0.874815i$$ $$L(\frac12)$$ $$\approx$$ $$0.212143 - 0.874815i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 1.73iT$$
7 $$1$$
good2 $$1 - T + 2T^{2}$$
5 $$1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 14T + 61T^{2}$$
67 $$1 - 4T + 67T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$