# Properties

 Label 2-882-7.4-c3-0-30 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 + (−1.72 − 2.98i)5-s − 7.99·8-s + (3.44 − 5.96i)10-s + (−18.0 + 31.2i)11-s − 10.2·13-s + (−8 − 13.8i)16-s + (59.2 − 102. i)17-s + (−19.3 − 33.4i)19-s + 13.7·20-s − 72.2·22-s + (18.1 + 31.3i)23-s + (56.5 − 97.9i)25-s + (−10.2 − 17.7i)26-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.154 − 0.266i)5-s − 0.353·8-s + (0.108 − 0.188i)10-s + (−0.495 + 0.857i)11-s − 0.218·13-s + (−0.125 − 0.216i)16-s + (0.845 − 1.46i)17-s + (−0.233 − 0.404i)19-s + 0.154·20-s − 0.700·22-s + (0.164 + 0.284i)23-s + (0.452 − 0.783i)25-s + (−0.0771 − 0.133i)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: $\chi_{882} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 882,\ (\ :3/2),\ 0.991 - 0.126i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.995679349$$ $$L(\frac12)$$ $$\approx$$ $$1.995679349$$ $$L(\frac{5}{2})$$ 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 + (1.72 + 2.98i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (18.0 - 31.2i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + 10.2T + 2.19e3T^{2}$$
17 $$1 + (-59.2 + 102. i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (19.3 + 33.4i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-18.1 - 31.3i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 12.1T + 2.43e4T^{2}$$
31 $$1 + (72.7 - 126. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (-0.685 - 1.18i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 168T + 6.89e4T^{2}$$
43 $$1 - 299.T + 7.95e4T^{2}$$
47 $$1 + (-251. - 435. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-312. + 541. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (21.1 - 36.5i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (219. + 380. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-381. + 661. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 1.02e3T + 3.57e5T^{2}$$
73 $$1 + (-289. + 501. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (471. + 816. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 474.T + 5.71e5T^{2}$$
89 $$1 + (-410. - 711. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 1.10e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$