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Properties

Label 2-882-7.4-c3-0-43
Degree $2$
Conductor $882$
Sign $0.749 + 0.661i$
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 + (3.70 + 6.42i)5-s − 7.99·8-s + (−7.41 + 12.8i)10-s + (5.24 − 9.08i)11-s + 2.78·13-s + (−8 − 13.8i)16-s + (25.2 − 43.6i)17-s + (−62.5 − 108. i)19-s − 29.6·20-s + 20.9·22-s + (−91.1 − 157. i)23-s + (35.0 − 60.6i)25-s + (2.78 + 4.82i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.331 + 0.574i)5-s − 0.353·8-s + (−0.234 + 0.406i)10-s + (0.143 − 0.248i)11-s + 0.0594·13-s + (−0.125 − 0.216i)16-s + (0.359 − 0.623i)17-s + (−0.754 − 1.30i)19-s − 0.331·20-s + 0.203·22-s + (−0.826 − 1.43i)23-s + (0.280 − 0.485i)25-s + (0.0210 + 0.0364i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
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: \((2,\ 882,\ (\ :3/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.605780356\)
\(L(\frac12)\) \(\approx\) \(1.605780356\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.70 - 6.42i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-5.24 + 9.08i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.78T + 2.19e3T^{2} \)
17 \( 1 + (-25.2 + 43.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (62.5 + 108. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (91.1 + 157. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 156.T + 2.43e4T^{2} \)
31 \( 1 + (69.8 - 120. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-197. - 341. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 197.T + 6.89e4T^{2} \)
43 \( 1 - 343.T + 7.95e4T^{2} \)
47 \( 1 + (305. + 528. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (68.7 - 119. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-294. + 510. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (123. + 214. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-197. + 342. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 285.T + 3.57e5T^{2} \)
73 \( 1 + (-498. + 863. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-424. - 734. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 210.T + 5.71e5T^{2} \)
89 \( 1 + (276. + 479. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 903.T + 9.12e5T^{2} \)
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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

−9.581689597120870241449029371330, −8.683618793309503247170240887829, −7.915846715995785561662580984003, −6.73361919148877356374227254377, −6.46949927214792239637585303120, −5.27449437201059194671885137545, −4.42067149340896726180764295017, −3.23177566150900069261710112847, −2.24409869367184915296051801650, −0.36588212544097549556219402180, 1.29437710147890435804708438873, 2.10522189486529417934661398755, 3.61427764118577519899886763179, 4.26045665583185082991370521831, 5.60004875572821065928934170306, 5.92684813783022966438984052132, 7.39834157230666578230040488177, 8.233517571829850689846109020501, 9.315766544681807959272304371303, 9.763407638025373129493629617875

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