# Properties

 Label 2-21e2-21.17-c3-0-31 Degree $2$ Conductor $441$ Sign $-0.851 + 0.524i$ Analytic cond. $26.0198$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.21 + 2.43i)2-s + (7.83 − 13.5i)4-s + (−6.38 − 11.0i)5-s + 37.3i·8-s + (53.7 + 31.0i)10-s + (−46.8 − 27.0i)11-s − 8.85i·13-s + (−28.1 − 48.7i)16-s + (34.4 − 59.6i)17-s + (141. − 81.9i)19-s − 200.·20-s + 263.·22-s + (81.3 − 46.9i)23-s + (−18.9 + 32.8i)25-s + (21.5 + 37.3i)26-s + ⋯
 L(s)  = 1 + (−1.48 + 0.860i)2-s + (0.979 − 1.69i)4-s + (−0.570 − 0.988i)5-s + 1.64i·8-s + (1.70 + 0.981i)10-s + (−1.28 − 0.741i)11-s − 0.188i·13-s + (−0.439 − 0.760i)16-s + (0.491 − 0.851i)17-s + (1.71 − 0.989i)19-s − 2.23·20-s + 2.55·22-s + (0.737 − 0.425i)23-s + (−0.151 + 0.262i)25-s + (0.162 + 0.281i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-0.851 + 0.524i$ Analytic conductor: $$26.0198$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{441} (80, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ -0.851 + 0.524i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.3330222936$$ $$L(\frac12)$$ $$\approx$$ $$0.3330222936$$ $$L(\frac{5}{2})$$ 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 + (4.21 - 2.43i)T + (4 - 6.92i)T^{2}$$
5 $$1 + (6.38 + 11.0i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (46.8 + 27.0i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + 8.85iT - 2.19e3T^{2}$$
17 $$1 + (-34.4 + 59.6i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-141. + 81.9i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-81.3 + 46.9i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 119. iT - 2.43e4T^{2}$$
31 $$1 + (85.6 + 49.4i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (47.0 + 81.5i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 259.T + 6.89e4T^{2}$$
43 $$1 - 5.01T + 7.95e4T^{2}$$
47 $$1 + (28.6 + 49.6i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-407. - 235. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-112. + 195. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (370. - 213. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (81.9 - 141. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 79.8iT - 3.57e5T^{2}$$
73 $$1 + (666. + 384. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (267. + 463. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 438.T + 5.71e5T^{2}$$
89 $$1 + (12.8 + 22.2i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 1.38e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$