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Properties

Label 2-882-7.2-c3-0-12
Degree $2$
Conductor $882$
Sign $-0.386 - 0.922i$
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 + (−3 + 5.19i)5-s + 7.99·8-s + (−6 − 10.3i)10-s + (−15 − 25.9i)11-s + 2·13-s + (−8 + 13.8i)16-s + (−33 − 57.1i)17-s + (26 − 45.0i)19-s + 24·20-s + 60·22-s + (−57 + 98.7i)23-s + (44.5 + 77.0i)25-s + (−2 + 3.46i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.268 + 0.464i)5-s + 0.353·8-s + (−0.189 − 0.328i)10-s + (−0.411 − 0.712i)11-s + 0.0426·13-s + (−0.125 + 0.216i)16-s + (−0.470 − 0.815i)17-s + (0.313 − 0.543i)19-s + 0.268·20-s + 0.581·22-s + (−0.516 + 0.895i)23-s + (0.355 + 0.616i)25-s + (−0.0150 + 0.0261i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
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: \((2,\ 882,\ (\ :3/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.053361001\)
\(L(\frac12)\) \(\approx\) \(1.053361001\)
\(L(\frac{5}{2})\) 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 \( 1 + (3 - 5.19i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15 + 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 2T + 2.19e3T^{2} \)
17 \( 1 + (33 + 57.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-26 + 45.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (57 - 98.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 72T + 2.43e4T^{2} \)
31 \( 1 + (-98 - 169. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-143 + 247. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 378T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 + (-114 + 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-174 - 301. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-174 - 301. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-53 + 91.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (298 + 516. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 630T + 3.57e5T^{2} \)
73 \( 1 + (-521 - 902. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-44 + 76.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.44e3T + 5.71e5T^{2} \)
89 \( 1 + (687 - 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 34T + 9.12e5T^{2} \)
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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

−9.902790238805984039776109117079, −9.057431468015248125979292916824, −8.303940316831252033972606853681, −7.36776001539196694753564458443, −6.79882935808722504087402662393, −5.71739176850383292411010324277, −4.92811501049603110385809074857, −3.64499683281847913923765425896, −2.56448878178511533615433797122, −0.908403316575332959191683866485, 0.39633018111258639945708327238, 1.71707677918271716372354227296, 2.78522023454477561243176007189, 4.10960330366790843041070634525, 4.72738928702710004218330272263, 6.01771529573056678937269654869, 7.04495507028091005635289317297, 8.240383328163199529541607929321, 8.421329719653034666935414173357, 9.740992415260192216760967640354

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