# Properties

 Label 2-21e2-21.20-c1-0-0 Degree $2$ Conductor $441$ Sign $0.716 - 0.698i$ Analytic cond. $3.52140$ Root an. cond. $1.87654$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.41i·2-s − 3.82·4-s − 3.37·5-s + 4.41i·8-s + 8.15i·10-s − 0.828i·11-s + 3.37i·13-s + 2.99·16-s − 1.39·17-s + 6.75i·19-s + 12.9·20-s − 1.99·22-s − 2i·23-s + 6.41·25-s + 8.15·26-s + ⋯
 L(s)  = 1 − 1.70i·2-s − 1.91·4-s − 1.51·5-s + 1.56i·8-s + 2.57i·10-s − 0.249i·11-s + 0.937i·13-s + 0.749·16-s − 0.339·17-s + 1.55i·19-s + 2.89·20-s − 0.426·22-s − 0.417i·23-s + 1.28·25-s + 1.59·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $0.716 - 0.698i$ Analytic conductor: $$3.52140$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{441} (440, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ 0.716 - 0.698i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.124213 + 0.0505284i$$ $$L(\frac12)$$ $$\approx$$ $$0.124213 + 0.0505284i$$ $$L(\frac{3}{2})$$ 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 + 2.41iT - 2T^{2}$$
5 $$1 + 3.37T + 5T^{2}$$
11 $$1 + 0.828iT - 11T^{2}$$
13 $$1 - 3.37iT - 13T^{2}$$
17 $$1 + 1.39T + 17T^{2}$$
19 $$1 - 6.75iT - 19T^{2}$$
23 $$1 + 2iT - 23T^{2}$$
29 $$1 - 4.82iT - 29T^{2}$$
31 $$1 + 6.75iT - 31T^{2}$$
37 $$1 + 2.58T + 37T^{2}$$
41 $$1 + 8.15T + 41T^{2}$$
43 $$1 + 12.4T + 43T^{2}$$
47 $$1 + 6.75T + 47T^{2}$$
53 $$1 - 7.07iT - 53T^{2}$$
59 $$1 + 6.75T + 59T^{2}$$
61 $$1 + 8.15iT - 61T^{2}$$
67 $$1 - 8.48T + 67T^{2}$$
71 $$1 + 4.82iT - 71T^{2}$$
73 $$1 + 1.39iT - 73T^{2}$$
79 $$1 + 9.65T + 79T^{2}$$
83 $$1 - 13.5T + 83T^{2}$$
89 $$1 + 6.17T + 89T^{2}$$
97 $$1 - 1.39iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$