Properties

 Label 2-882-7.4-c3-0-12 Degree $2$ Conductor $882$ Sign $0.968 + 0.250i$ Analytic cond. $52.0396$ Root an. cond. $7.21385$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (−11 − 19.0i)5-s + 7.99·8-s + (−22 + 38.1i)10-s + (13 − 22.5i)11-s + 54·13-s + (−8 − 13.8i)16-s + (−37 + 64.0i)17-s + (58 + 100. i)19-s + 88·20-s − 51.9·22-s + (−29 − 50.2i)23-s + (−179.5 + 310. i)25-s + (−54 − 93.5i)26-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.983 − 1.70i)5-s + 0.353·8-s + (−0.695 + 1.20i)10-s + (0.356 − 0.617i)11-s + 1.15·13-s + (−0.125 − 0.216i)16-s + (−0.527 + 0.914i)17-s + (0.700 + 1.21i)19-s + 0.983·20-s − 0.503·22-s + (−0.262 − 0.455i)23-s + (−1.43 + 2.48i)25-s + (−0.407 − 0.705i)26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$882$$    =    $$2 \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.968 + 0.250i$ Analytic conductor: $$52.0396$$ Root analytic conductor: $$7.21385$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{882} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 882,\ (\ :3/2),\ 0.968 + 0.250i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.9782505264$$ $$L(\frac12)$$ $$\approx$$ $$0.9782505264$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1 + 1.73i)T$$
3 $$1$$
7 $$1$$
good5 $$1 + (11 + 19.0i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-13 + 22.5i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 54T + 2.19e3T^{2}$$
17 $$1 + (37 - 64.0i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-58 - 100. i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (29 + 50.2i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 208T + 2.43e4T^{2}$$
31 $$1 + (126 - 218. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (25 + 43.3i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 - 126T + 6.89e4T^{2}$$
43 $$1 - 164T + 7.95e4T^{2}$$
47 $$1 + (-222 - 384. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-6 + 10.3i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (62 - 107. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (81 + 140. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-430 + 744. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 238T + 3.57e5T^{2}$$
73 $$1 + (73 - 126. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-492 - 852. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 656T + 5.71e5T^{2}$$
89 $$1 + (-477 - 826. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 526T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$