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Properties

Label 2-882-7.4-c3-0-21
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 $0$

Origins

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Normalization:  

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

\(L(2)\) \(\approx\) \(2.990672975\)
\(L(\frac12)\) \(\approx\) \(2.990672975\)
\(L(\frac{5}{2})\) 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 \( 1 + 526T + 9.12e5T^{2} \)
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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

−10.10324970872910118563993969350, −9.472611466745523857975637391098, −8.246969507104555018800315631606, −7.25548772435266672522130566870, −6.75584249299222292181317890609, −5.85836316998607796306946506728, −5.18868273799001947003019907772, −3.58646644756029227980934631957, −2.96086020034435439386371356470, −1.65304744305588622253064255264, 0.72073454422683459054382020477, 1.40803933143798587166975529332, 2.68802811088496267993818996845, 4.00994707850092303655913556912, 4.92954993586069015402247425827, 5.67300117162461797580523395188, 6.33608442100946324051894453340, 8.056738954645379092987383435277, 8.719982553543262981811542874834, 9.337004952024934762886630898959

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