# Properties

 Label 2-21e2-63.58-c1-0-15 Degree $2$ Conductor $441$ Sign $0.467 + 0.883i$ 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 + 0.670·2-s + (−1.65 − 0.518i)3-s − 1.55·4-s + (0.712 + 1.23i)5-s + (−1.10 − 0.347i)6-s − 2.38·8-s + (2.46 + 1.71i)9-s + (0.477 + 0.827i)10-s + (2.46 − 4.27i)11-s + (2.56 + 0.803i)12-s + (1.37 − 2.38i)13-s + (−0.537 − 2.40i)15-s + 1.50·16-s + (−0.559 − 0.969i)17-s + (1.65 + 1.14i)18-s + (2.00 − 3.47i)19-s + ⋯
 L(s)  = 1 + 0.473·2-s + (−0.954 − 0.299i)3-s − 0.775·4-s + (0.318 + 0.551i)5-s + (−0.452 − 0.141i)6-s − 0.841·8-s + (0.820 + 0.571i)9-s + (0.151 + 0.261i)10-s + (0.743 − 1.28i)11-s + (0.739 + 0.232i)12-s + (0.381 − 0.661i)13-s + (−0.138 − 0.621i)15-s + 0.376·16-s + (−0.135 − 0.235i)17-s + (0.389 + 0.270i)18-s + (0.460 − 0.797i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $0.467 + 0.883i$ Analytic conductor: $$3.52140$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{441} (373, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ 0.467 + 0.883i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.899510 - 0.541532i$$ $$L(\frac12)$$ $$\approx$$ $$0.899510 - 0.541532i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.65 + 0.518i)T$$
7 $$1$$
good2 $$1 - 0.670T + 2T^{2}$$
5 $$1 + (-0.712 - 1.23i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-2.46 + 4.27i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-1.37 + 2.38i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (0.559 + 0.969i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (2.71 + 4.70i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 2.50T + 31T^{2}$$
37 $$1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 - 9.47T + 47T^{2}$$
53 $$1 + (0.410 + 0.710i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 - 6.58T + 59T^{2}$$
61 $$1 + 0.0752T + 61T^{2}$$
67 $$1 + 12.5T + 67T^{2}$$
71 $$1 - 0.0804T + 71T^{2}$$
73 $$1 + (5.34 + 9.25i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + 1.84T + 79T^{2}$$
83 $$1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (6.76 - 11.7i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (2.70 + 4.67i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$