# Properties

 Label 2-21e2-9.4-c1-0-24 Degree $2$ Conductor $441$ Sign $-1$ 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 + (−0.673 − 1.16i)2-s + (−1.70 + 0.300i)3-s + (0.0923 − 0.160i)4-s + (1.26 − 2.19i)5-s + (1.49 + 1.78i)6-s − 2.94·8-s + (2.81 − 1.02i)9-s − 3.41·10-s + (−0.233 − 0.405i)11-s + (−0.109 + 0.300i)12-s + (2.91 − 5.04i)13-s + (−1.5 + 4.12i)15-s + (1.79 + 3.11i)16-s − 3.87·17-s + (−3.09 − 2.59i)18-s + 2.18·19-s + ⋯
 L(s)  = 1 + (−0.476 − 0.825i)2-s + (−0.984 + 0.173i)3-s + (0.0461 − 0.0800i)4-s + (0.566 − 0.980i)5-s + (0.612 + 0.729i)6-s − 1.04·8-s + (0.939 − 0.342i)9-s − 1.07·10-s + (−0.0705 − 0.122i)11-s + (−0.0316 + 0.0868i)12-s + (0.807 − 1.39i)13-s + (−0.387 + 1.06i)15-s + (0.449 + 0.778i)16-s − 0.940·17-s + (−0.729 − 0.612i)18-s + 0.501·19-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$3.52140$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{441} (148, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.706057i$$ $$L(\frac12)$$ $$\approx$$ $$0.706057i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.70 - 0.300i)T$$
7 $$1$$
good2 $$1 + (0.673 + 1.16i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (0.233 + 0.405i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (-2.91 + 5.04i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + 3.87T + 17T^{2}$$
19 $$1 - 2.18T + 19T^{2}$$
23 $$1 + (-0.0530 + 0.0918i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (3.84 - 6.65i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + 7.68T + 37T^{2}$$
41 $$1 + (1.11 - 1.92i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (2.66 + 4.61i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 0.716T + 53T^{2}$$
59 $$1 + (-0.368 + 0.637i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-0.479 - 0.829i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-4.81 + 8.34i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 13.2T + 71T^{2}$$
73 $$1 - 10.2T + 73T^{2}$$
79 $$1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 8.11T + 89T^{2}$$
97 $$1 + (6.80 + 11.7i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$