# Properties

 Label 2-21e2-7.2-c3-0-16 Degree $2$ Conductor $441$ Sign $-0.198 - 0.980i$ 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 + (−2.69 + 4.67i)2-s + (−10.5 − 18.3i)4-s + (−7.78 + 13.4i)5-s + 71.0·8-s + (−42.0 − 72.8i)10-s + (−15.9 − 27.6i)11-s − 72.5·13-s + (−107. + 185. i)16-s + (14.5 + 25.1i)17-s + (54.4 − 94.2i)19-s + 329.·20-s + 172.·22-s + (27.6 − 47.8i)23-s + (−58.8 − 101. i)25-s + (195. − 339. i)26-s + ⋯
 L(s)  = 1 + (−0.954 + 1.65i)2-s + (−1.32 − 2.29i)4-s + (−0.696 + 1.20i)5-s + 3.14·8-s + (−1.32 − 2.30i)10-s + (−0.438 − 0.758i)11-s − 1.54·13-s + (−1.67 + 2.90i)16-s + (0.207 + 0.358i)17-s + (0.657 − 1.13i)19-s + 3.68·20-s + 1.67·22-s + (0.250 − 0.434i)23-s + (−0.470 − 0.815i)25-s + (1.47 − 2.56i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-0.198 - 0.980i$ Analytic conductor: $$26.0198$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{441} (226, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ -0.198 - 0.980i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.5879015858$$ $$L(\frac12)$$ $$\approx$$ $$0.5879015858$$ $$L(\frac{5}{2})$$ 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 + (2.69 - 4.67i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (7.78 - 13.4i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (15.9 + 27.6i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 72.5T + 2.19e3T^{2}$$
17 $$1 + (-14.5 - 25.1i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-54.4 + 94.2i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-27.6 + 47.8i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 17.7T + 2.43e4T^{2}$$
31 $$1 + (-28.0 - 48.6i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-147. + 256. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 238.T + 6.89e4T^{2}$$
43 $$1 - 16.8T + 7.95e4T^{2}$$
47 $$1 + (255. - 443. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (132. + 229. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-127. - 220. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (36.4 - 63.0i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-253. - 438. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 827.T + 3.57e5T^{2}$$
73 $$1 + (186. + 322. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (514. - 890. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 453.T + 5.71e5T^{2}$$
89 $$1 + (166. - 287. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 1.16e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$