# Properties

 Label 2-882-7.4-c3-0-27 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 + (7.5 + 12.9i)5-s + 7.99·8-s + (15 − 25.9i)10-s + (−4.5 + 7.79i)11-s + 88·13-s + (−8 − 13.8i)16-s + (42 − 72.7i)17-s + (52 + 90.0i)19-s − 60·20-s + 18·22-s + (−42 − 72.7i)23-s + (−50 + 86.6i)25-s + (−88 − 152. i)26-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + 0.353·8-s + (0.474 − 0.821i)10-s + (−0.123 + 0.213i)11-s + 1.87·13-s + (−0.125 − 0.216i)16-s + (0.599 − 1.03i)17-s + (0.627 + 1.08i)19-s − 0.670·20-s + 0.174·22-s + (−0.380 − 0.659i)23-s + (−0.400 + 0.692i)25-s + (−0.663 − 1.14i)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

 $$L(2)$$ $$\approx$$ $$2.201802374$$ $$L(\frac12)$$ $$\approx$$ $$2.201802374$$ $$L(\frac{5}{2})$$ 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 + (-7.5 - 12.9i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (4.5 - 7.79i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 88T + 2.19e3T^{2}$$
17 $$1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-52 - 90.0i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (42 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 51T + 2.43e4T^{2}$$
31 $$1 + (-92.5 + 160. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (22 + 38.1i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 168T + 6.89e4T^{2}$$
43 $$1 - 326T + 7.95e4T^{2}$$
47 $$1 + (-69 - 119. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-319.5 + 553. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (79.5 - 137. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-361 - 625. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-83 + 143. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 1.08e3T + 3.57e5T^{2}$$
73 $$1 + (-109 + 188. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-291.5 - 504. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 597T + 5.71e5T^{2}$$
89 $$1 + (-519 - 898. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 169T + 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.03245559958017064118179415095, −9.108405651408045119036910910646, −8.152505378630798299159528353568, −7.28174100178960210009641039443, −6.28576223143553926093655555174, −5.57183141114174089979452010651, −4.01945838171820216718624890850, −3.16705677521830510990103354373, −2.20253531835108513668716009281, −0.989890241946753010844531822044, 0.871371088139521781582173889999, 1.61099360543715266666972509201, 3.43342528532717521611183859716, 4.59365991708199108903706476370, 5.64530164833488729003380350719, 6.02214964050695360950919815442, 7.21621616189477991160094011090, 8.352408264812863941792285192478, 8.726937585838005903889681790942, 9.487197656409406714988870145216