Properties

 Degree $2$ Conductor $441$ Sign $-0.266 - 0.963i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1.5 − 2.59i)2-s + (−0.5 + 0.866i)4-s + (9 + 15.5i)5-s − 21·8-s + (27 − 46.7i)10-s + (−18 + 31.1i)11-s + 34·13-s + (35.5 + 61.4i)16-s + (−21 + 36.3i)17-s + (−62 − 107. i)19-s − 18.0·20-s + 108·22-s + (−99.5 + 172. i)25-s + (−51 − 88.3i)26-s − 102·29-s + ⋯
 L(s)  = 1 + (−0.530 − 0.918i)2-s + (−0.0625 + 0.108i)4-s + (0.804 + 1.39i)5-s − 0.928·8-s + (0.853 − 1.47i)10-s + (−0.493 + 0.854i)11-s + 0.725·13-s + (0.554 + 0.960i)16-s + (−0.299 + 0.518i)17-s + (−0.748 − 1.29i)19-s − 0.201·20-s + 1.04·22-s + (−0.796 + 1.37i)25-s + (−0.384 − 0.666i)26-s − 0.653·29-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-0.266 - 0.963i$ Motivic weight: $$3$$ Character: $\chi_{441} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ -0.266 - 0.963i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.5612293397$$ $$L(\frac12)$$ $$\approx$$ $$0.5612293397$$ $$L(\frac{5}{2})$$ 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 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2}$$
5 $$1 + (-9 - 15.5i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 34T + 2.19e3T^{2}$$
17 $$1 + (21 - 36.3i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (62 + 107. i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 102T + 2.43e4T^{2}$$
31 $$1 + (80 - 138. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (199 + 344. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 318T + 6.89e4T^{2}$$
43 $$1 + 268T + 7.95e4T^{2}$$
47 $$1 + (120 + 207. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (249 - 431. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-66 + 114. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-199 - 344. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (46 - 79.6i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 720T + 3.57e5T^{2}$$
73 $$1 + (251 - 434. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-512 - 886. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 204T + 5.71e5T^{2}$$
89 $$1 + (177 + 306. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 286T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$