# Properties

 Label 2-882-7.2-c3-0-6 Degree $2$ Conductor $882$ Sign $-0.991 - 0.126i$ 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 + (−5.22 + 9.04i)5-s + 7.99·8-s + (−10.4 − 18.0i)10-s + (−30.5 − 52.9i)11-s + 59.2·13-s + (−8 + 13.8i)16-s + (10.2 + 17.7i)17-s + (−40.1 + 69.5i)19-s + 41.7·20-s + 122.·22-s + (79.1 − 137. i)23-s + (7.93 + 13.7i)25-s + (−59.2 + 102. i)26-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.467 + 0.809i)5-s + 0.353·8-s + (−0.330 − 0.572i)10-s + (−0.837 − 1.45i)11-s + 1.26·13-s + (−0.125 + 0.216i)16-s + (0.145 + 0.252i)17-s + (−0.485 + 0.840i)19-s + 0.467·20-s + 1.18·22-s + (0.717 − 1.24i)23-s + (0.0635 + 0.109i)25-s + (−0.446 + 0.773i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$882$$    =    $$2 \cdot 3^{2} \cdot 7^{2}$$ Sign: $-0.991 - 0.126i$ Analytic conductor: $$52.0396$$ Root analytic conductor: $$7.21385$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{882} (667, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 882,\ (\ :3/2),\ -0.991 - 0.126i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.6863404268$$ $$L(\frac12)$$ $$\approx$$ $$0.6863404268$$ $$L(\frac{5}{2})$$ 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 + (5.22 - 9.04i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (30.5 + 52.9i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 - 59.2T + 2.19e3T^{2}$$
17 $$1 + (-10.2 - 17.7i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (40.1 - 69.5i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-79.1 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 85.1T + 2.43e4T^{2}$$
31 $$1 + (-121. - 210. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (145. - 251. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 168T + 6.89e4T^{2}$$
43 $$1 - 7.62T + 7.95e4T^{2}$$
47 $$1 + (84.6 - 146. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-125. - 216. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (402. + 697. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-16.5 + 28.7i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-138. - 240. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 631.T + 3.57e5T^{2}$$
73 $$1 + (384. + 665. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-209. + 362. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 761.T + 5.71e5T^{2}$$
89 $$1 + (786. - 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 1.04e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$