# Properties

 Label 2-21e2-1.1-c3-0-33 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 + 4.55·2-s + 12.7·4-s + 17.8·5-s + 21.6·8-s + 81.4·10-s + 11.3·11-s + 13.0·13-s − 3.25·16-s + 53.2·17-s + 42.4·19-s + 228.·20-s + 51.9·22-s − 152.·23-s + 194.·25-s + 59.6·26-s − 186.·29-s + 157.·31-s − 188.·32-s + 242.·34-s + 3.74·37-s + 193.·38-s + 387.·40-s − 39.3·41-s + 429.·43-s + 145.·44-s − 692.·46-s + 21.1·47-s + ⋯
 L(s)  = 1 + 1.61·2-s + 1.59·4-s + 1.59·5-s + 0.958·8-s + 2.57·10-s + 0.312·11-s + 0.279·13-s − 0.0508·16-s + 0.759·17-s + 0.512·19-s + 2.54·20-s + 0.503·22-s − 1.37·23-s + 1.55·25-s + 0.450·26-s − 1.19·29-s + 0.914·31-s − 1.04·32-s + 1.22·34-s + 0.0166·37-s + 0.825·38-s + 1.53·40-s − 0.149·41-s + 1.52·43-s + 0.498·44-s − 2.22·46-s + 0.0657·47-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

 $$L(2)$$ $$\approx$$ $$6.407208894$$ $$L(\frac12)$$ $$\approx$$ $$6.407208894$$ $$L(\frac{5}{2})$$ 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 - 4.55T + 8T^{2}$$
5 $$1 - 17.8T + 125T^{2}$$
11 $$1 - 11.3T + 1.33e3T^{2}$$
13 $$1 - 13.0T + 2.19e3T^{2}$$
17 $$1 - 53.2T + 4.91e3T^{2}$$
19 $$1 - 42.4T + 6.85e3T^{2}$$
23 $$1 + 152.T + 1.21e4T^{2}$$
29 $$1 + 186.T + 2.43e4T^{2}$$
31 $$1 - 157.T + 2.97e4T^{2}$$
37 $$1 - 3.74T + 5.06e4T^{2}$$
41 $$1 + 39.3T + 6.89e4T^{2}$$
43 $$1 - 429.T + 7.95e4T^{2}$$
47 $$1 - 21.1T + 1.03e5T^{2}$$
53 $$1 + 365.T + 1.48e5T^{2}$$
59 $$1 + 226.T + 2.05e5T^{2}$$
61 $$1 + 651.T + 2.26e5T^{2}$$
67 $$1 - 145.T + 3.00e5T^{2}$$
71 $$1 - 368.T + 3.57e5T^{2}$$
73 $$1 + 608.T + 3.89e5T^{2}$$
79 $$1 - 910.T + 4.93e5T^{2}$$
83 $$1 + 327.T + 5.71e5T^{2}$$
89 $$1 + 37.6T + 7.04e5T^{2}$$
97 $$1 + 722.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$