# Properties

 Degree $2$ Conductor $441$ Sign $0.266 + 0.963i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (1 + 1.73i)5-s − 3·8-s + (0.999 − 1.73i)10-s + (2 − 3.46i)11-s + 2·13-s + (0.500 + 0.866i)16-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + 2·20-s − 3.99·22-s + (0.500 − 0.866i)25-s + (−1 − 1.73i)26-s + 2·29-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (0.447 + 0.774i)5-s − 1.06·8-s + (0.316 − 0.547i)10-s + (0.603 − 1.04i)11-s + 0.554·13-s + (0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + 0.447·20-s − 0.852·22-s + (0.100 − 0.173i)25-s + (−0.196 − 0.339i)26-s + 0.371·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $0.266 + 0.963i$ Motivic weight: $$1$$ Character: $\chi_{441} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ 0.266 + 0.963i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.08560 - 0.825880i$$ $$L(\frac12)$$ $$\approx$$ $$1.08560 - 0.825880i$$ $$L(\frac{3}{2})$$ 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 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 12T + 83T^{2}$$
89 $$1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 18T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$