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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 $0$

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line