# Properties

 Degree $2$ Conductor $441$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.95·2-s − 4.15·4-s − 11.8·5-s − 23.8·8-s − 23.3·10-s + 36.4·11-s − 0.964·13-s − 13.4·16-s − 98.2·17-s + 106.·19-s + 49.4·20-s + 71.3·22-s + 54.3·23-s + 16.3·25-s − 1.89·26-s + 229.·29-s − 127.·31-s + 164.·32-s − 192.·34-s + 311.·37-s + 207.·38-s + 283.·40-s + 419.·41-s + 523.·43-s − 151.·44-s + 106.·46-s − 270.·47-s + ⋯
 L(s)  = 1 + 0.692·2-s − 0.519·4-s − 1.06·5-s − 1.05·8-s − 0.736·10-s + 0.998·11-s − 0.0205·13-s − 0.209·16-s − 1.40·17-s + 1.27·19-s + 0.552·20-s + 0.691·22-s + 0.492·23-s + 0.130·25-s − 0.0142·26-s + 1.47·29-s − 0.740·31-s + 0.907·32-s − 0.971·34-s + 1.38·37-s + 0.886·38-s + 1.11·40-s + 1.59·41-s + 1.85·43-s − 0.518·44-s + 0.341·46-s − 0.838·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$3$$ Character: $\chi_{441} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.674358649$$ $$L(\frac12)$$ $$\approx$$ $$1.674358649$$ $$L(\frac{5}{2})$$ 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 - 1.95T + 8T^{2}$$
5 $$1 + 11.8T + 125T^{2}$$
11 $$1 - 36.4T + 1.33e3T^{2}$$
13 $$1 + 0.964T + 2.19e3T^{2}$$
17 $$1 + 98.2T + 4.91e3T^{2}$$
19 $$1 - 106.T + 6.85e3T^{2}$$
23 $$1 - 54.3T + 1.21e4T^{2}$$
29 $$1 - 229.T + 2.43e4T^{2}$$
31 $$1 + 127.T + 2.97e4T^{2}$$
37 $$1 - 311.T + 5.06e4T^{2}$$
41 $$1 - 419.T + 6.89e4T^{2}$$
43 $$1 - 523.T + 7.95e4T^{2}$$
47 $$1 + 270.T + 1.03e5T^{2}$$
53 $$1 + 251.T + 1.48e5T^{2}$$
59 $$1 + 408.T + 2.05e5T^{2}$$
61 $$1 - 860.T + 2.26e5T^{2}$$
67 $$1 - 506.T + 3.00e5T^{2}$$
71 $$1 + 523.T + 3.57e5T^{2}$$
73 $$1 + 629.T + 3.89e5T^{2}$$
79 $$1 - 319.T + 4.93e5T^{2}$$
83 $$1 + 1.30e3T + 5.71e5T^{2}$$
89 $$1 - 348.T + 7.04e5T^{2}$$
97 $$1 - 161.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$