# Properties

 Label 2-21e2-1.1-c3-0-41 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 + 2.64·2-s − 0.999·4-s − 23.8·8-s + 26.4·11-s − 55.0·16-s + 70·22-s − 216.·23-s − 125·25-s − 264.·29-s + 44.9·32-s − 450·37-s + 180·43-s − 26.4·44-s − 574·46-s − 330.·50-s + 497.·53-s − 700.·58-s + 559·64-s − 740·67-s + 978.·71-s − 1.19e3·74-s − 1.38e3·79-s + 476.·86-s − 630·88-s + 216.·92-s + 124.·100-s + 1.31e3·106-s + ⋯
 L(s)  = 1 + 0.935·2-s − 0.124·4-s − 1.05·8-s + 0.725·11-s − 0.859·16-s + 0.678·22-s − 1.96·23-s − 25-s − 1.69·29-s + 0.248·32-s − 1.99·37-s + 0.638·43-s − 0.0906·44-s − 1.83·46-s − 0.935·50-s + 1.28·53-s − 1.58·58-s + 1.09·64-s − 1.34·67-s + 1.63·71-s − 1.87·74-s − 1.97·79-s + 0.597·86-s − 0.763·88-s + 0.245·92-s + 0.124·100-s + 1.20·106-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 - 2.64T + 8T^{2}$$
5 $$1 + 125T^{2}$$
11 $$1 - 26.4T + 1.33e3T^{2}$$
13 $$1 + 2.19e3T^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 + 6.85e3T^{2}$$
23 $$1 + 216.T + 1.21e4T^{2}$$
29 $$1 + 264.T + 2.43e4T^{2}$$
31 $$1 + 2.97e4T^{2}$$
37 $$1 + 450T + 5.06e4T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 180T + 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 497.T + 1.48e5T^{2}$$
59 $$1 + 2.05e5T^{2}$$
61 $$1 + 2.26e5T^{2}$$
67 $$1 + 740T + 3.00e5T^{2}$$
71 $$1 - 978.T + 3.57e5T^{2}$$
73 $$1 + 3.89e5T^{2}$$
79 $$1 + 1.38e3T + 4.93e5T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$