# Properties

 Label 2-21e2-1.1-c3-0-44 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.53·2-s + 4.46·4-s − 2.07·5-s − 12.4·8-s − 7.33·10-s − 49.1·11-s + 44.8·13-s − 79.7·16-s + 26.5·17-s − 77.7·19-s − 9.28·20-s − 173.·22-s − 55.7·23-s − 120.·25-s + 158.·26-s − 121.·29-s − 305.·31-s − 181.·32-s + 93.6·34-s + 77.1·37-s − 274.·38-s + 25.9·40-s + 248.·41-s − 147.·43-s − 219.·44-s − 196.·46-s + 269.·47-s + ⋯
 L(s)  = 1 + 1.24·2-s + 0.558·4-s − 0.185·5-s − 0.551·8-s − 0.231·10-s − 1.34·11-s + 0.956·13-s − 1.24·16-s + 0.378·17-s − 0.938·19-s − 0.103·20-s − 1.68·22-s − 0.505·23-s − 0.965·25-s + 1.19·26-s − 0.777·29-s − 1.77·31-s − 1.00·32-s + 0.472·34-s + 0.342·37-s − 1.17·38-s + 0.102·40-s + 0.947·41-s − 0.521·43-s − 0.753·44-s − 0.630·46-s + 0.837·47-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: $$1$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 3.53T + 8T^{2}$$
5 $$1 + 2.07T + 125T^{2}$$
11 $$1 + 49.1T + 1.33e3T^{2}$$
13 $$1 - 44.8T + 2.19e3T^{2}$$
17 $$1 - 26.5T + 4.91e3T^{2}$$
19 $$1 + 77.7T + 6.85e3T^{2}$$
23 $$1 + 55.7T + 1.21e4T^{2}$$
29 $$1 + 121.T + 2.43e4T^{2}$$
31 $$1 + 305.T + 2.97e4T^{2}$$
37 $$1 - 77.1T + 5.06e4T^{2}$$
41 $$1 - 248.T + 6.89e4T^{2}$$
43 $$1 + 147.T + 7.95e4T^{2}$$
47 $$1 - 269.T + 1.03e5T^{2}$$
53 $$1 - 141.T + 1.48e5T^{2}$$
59 $$1 + 424.T + 2.05e5T^{2}$$
61 $$1 - 587.T + 2.26e5T^{2}$$
67 $$1 + 179.T + 3.00e5T^{2}$$
71 $$1 + 674.T + 3.57e5T^{2}$$
73 $$1 + 237.T + 3.89e5T^{2}$$
79 $$1 - 495.T + 4.93e5T^{2}$$
83 $$1 + 24.4T + 5.71e5T^{2}$$
89 $$1 - 1.07e3T + 7.04e5T^{2}$$
97 $$1 + 1.66e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$