# Properties

 Label 2-21e2-1.1-c3-0-4 Degree $2$ Conductor $441$ Sign $1$ Analytic cond. $26.0198$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4.05·2-s + 8.44·4-s − 9.92·5-s − 1.80·8-s + 40.2·10-s − 13.5·11-s + 18.5·13-s − 60.2·16-s − 93.7·17-s + 131.·19-s − 83.7·20-s + 54.9·22-s − 198.·23-s − 26.5·25-s − 75.2·26-s + 188.·29-s − 83.9·31-s + 258.·32-s + 380.·34-s + 80.1·37-s − 534.·38-s + 17.9·40-s − 385.·41-s − 397.·43-s − 114.·44-s + 804.·46-s + 272.·47-s + ⋯
 L(s)  = 1 − 1.43·2-s + 1.05·4-s − 0.887·5-s − 0.0799·8-s + 1.27·10-s − 0.371·11-s + 0.395·13-s − 0.941·16-s − 1.33·17-s + 1.59·19-s − 0.936·20-s + 0.532·22-s − 1.79·23-s − 0.212·25-s − 0.567·26-s + 1.20·29-s − 0.486·31-s + 1.42·32-s + 1.91·34-s + 0.356·37-s − 2.28·38-s + 0.0709·40-s − 1.46·41-s − 1.40·43-s − 0.391·44-s + 2.57·46-s + 0.845·47-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: $1$ Analytic conductor: $$26.0198$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{441} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.4901751783$$ $$L(\frac12)$$ $$\approx$$ $$0.4901751783$$ $$L(\frac{5}{2})$$ 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 + 4.05T + 8T^{2}$$
5 $$1 + 9.92T + 125T^{2}$$
11 $$1 + 13.5T + 1.33e3T^{2}$$
13 $$1 - 18.5T + 2.19e3T^{2}$$
17 $$1 + 93.7T + 4.91e3T^{2}$$
19 $$1 - 131.T + 6.85e3T^{2}$$
23 $$1 + 198.T + 1.21e4T^{2}$$
29 $$1 - 188.T + 2.43e4T^{2}$$
31 $$1 + 83.9T + 2.97e4T^{2}$$
37 $$1 - 80.1T + 5.06e4T^{2}$$
41 $$1 + 385.T + 6.89e4T^{2}$$
43 $$1 + 397.T + 7.95e4T^{2}$$
47 $$1 - 272.T + 1.03e5T^{2}$$
53 $$1 + 36.9T + 1.48e5T^{2}$$
59 $$1 - 395.T + 2.05e5T^{2}$$
61 $$1 - 13.4T + 2.26e5T^{2}$$
67 $$1 - 340.T + 3.00e5T^{2}$$
71 $$1 + 211.T + 3.57e5T^{2}$$
73 $$1 - 486.T + 3.89e5T^{2}$$
79 $$1 - 293.T + 4.93e5T^{2}$$
83 $$1 - 889.T + 5.71e5T^{2}$$
89 $$1 - 1.14e3T + 7.04e5T^{2}$$
97 $$1 - 1.38e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$