# Properties

 Degree $2$ Conductor $441$ Sign $-0.198 - 0.980i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.207 − 0.358i)2-s + (0.914 + 1.58i)4-s + (−1.70 + 2.95i)5-s + 1.58·8-s + (0.707 + 1.22i)10-s + (−1 − 1.73i)11-s − 2.58·13-s + (−1.49 + 2.59i)16-s + (1.12 + 1.94i)17-s + (−1.41 + 2.44i)19-s − 6.24·20-s − 0.828·22-s + (−3.82 + 6.63i)23-s + (−3.32 − 5.76i)25-s + (−0.535 + 0.927i)26-s + ⋯
 L(s)  = 1 + (0.146 − 0.253i)2-s + (0.457 + 0.791i)4-s + (−0.763 + 1.32i)5-s + 0.560·8-s + (0.223 + 0.387i)10-s + (−0.301 − 0.522i)11-s − 0.717·13-s + (−0.374 + 0.649i)16-s + (0.271 + 0.471i)17-s + (−0.324 + 0.561i)19-s − 1.39·20-s − 0.176·22-s + (−0.798 + 1.38i)23-s + (−0.665 − 1.15i)25-s + (−0.105 + 0.181i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-0.198 - 0.980i$ Motivic weight: $$1$$ Character: $\chi_{441} (226, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ -0.198 - 0.980i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.797675 + 0.974976i$$ $$L(\frac12)$$ $$\approx$$ $$0.797675 + 0.974976i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + (-0.207 + 0.358i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + 2.58T + 13T^{2}$$
17 $$1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (3.82 - 6.63i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 - 6.82T + 29T^{2}$$
31 $$1 + (0.585 + 1.01i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 6.24T + 41T^{2}$$
43 $$1 - 5.65T + 43T^{2}$$
47 $$1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-0.585 - 1.01i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-6.12 + 10.6i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 9.31T + 71T^{2}$$
73 $$1 + (-6.94 - 12.0i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 7.31T + 83T^{2}$$
89 $$1 + (-7.12 + 12.3i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 2.58T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$