Properties

Label 2-882-7.4-c3-0-12
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

\(L(2)\) \(\approx\) \(0.9782505264\)
\(L(\frac12)\) \(\approx\) \(0.9782505264\)
\(L(\frac{5}{2})\) 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

−9.432825958910248242204933853110, −8.878316133808982505581640176564, −8.248814367498077897165265047481, −7.63091023140168446810119532475, −6.09380494144467506112793324923, −5.17091914651107772065302163293, −3.92162006888989307190914822155, −3.68054625990379606434947005291, −1.63873869919588307507488588616, −0.887657582814750974038995961043, 0.39013114555861141962453498791, 2.26748573469788549057938807006, 3.46918096519184009751037377914, 4.27028959715266063576473322013, 5.67838750235509935307533007505, 6.65411016743907223561366470271, 7.25703577868893716890518969158, 7.74052868467218168454395702554, 8.948596996931048444671945232893, 9.679307336427819399674734133445

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