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Properties

Label 2-882-7.4-c3-0-39
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 + (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

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

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: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.035999508\)
\(L(\frac12)\) \(\approx\) \(1.035999508\)
\(L(\frac{5}{2})\) 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

−9.773323800750509909203951255043, −8.971298422553563290256116191898, −7.65213484358953468906103494392, −6.97080313401990448693295940294, −6.16871000485566047389421838542, −5.08595911866038596763805950688, −3.65087541392379821884583597952, −2.66996326035878123240456937164, −2.12032342791897659795775838008, −0.29578349224155031315273062569, 1.26563670048215610476525433481, 1.98744051570252632130547861794, 4.04695018025439706339452306554, 4.94728763631310851907447983212, 5.63303940277525784098928761012, 6.43824423174904754658920166588, 7.67911354794549157461584547997, 8.367528446415071956087657778688, 9.166955206717308768430877443598, 9.843893248102421437300247120544

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