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Properties

Label 2-882-3.2-c4-0-44
Degree $2$
Conductor $882$
Sign $0.816 + 0.577i$
Analytic cond. $91.1723$
Root an. cond. $9.54841$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 8.00·4-s − 11.5i·5-s − 22.6i·8-s + 32.6·10-s − 212. i·11-s + 288.·13-s + 64.0·16-s + 397. i·17-s − 346.·19-s + 92.2i·20-s + 601.·22-s + 659. i·23-s + 491.·25-s + 815. i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s − 0.461i·5-s − 0.353i·8-s + 0.326·10-s − 1.75i·11-s + 1.70·13-s + 0.250·16-s + 1.37i·17-s − 0.960·19-s + 0.230i·20-s + 1.24·22-s + 1.24i·23-s + 0.787·25-s + 1.20i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(91.1723\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :2),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.863687588\)
\(L(\frac12)\) \(\approx\) \(1.863687588\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 11.5iT - 625T^{2} \)
11 \( 1 + 212. iT - 1.46e4T^{2} \)
13 \( 1 - 288.T + 2.85e4T^{2} \)
17 \( 1 - 397. iT - 8.35e4T^{2} \)
19 \( 1 + 346.T + 1.30e5T^{2} \)
23 \( 1 - 659. iT - 2.79e5T^{2} \)
29 \( 1 + 192. iT - 7.07e5T^{2} \)
31 \( 1 - 613.T + 9.23e5T^{2} \)
37 \( 1 + 2.17e3T + 1.87e6T^{2} \)
41 \( 1 + 2.35e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.25e3T + 3.41e6T^{2} \)
47 \( 1 + 3.12e3iT - 4.87e6T^{2} \)
53 \( 1 - 112. iT - 7.89e6T^{2} \)
59 \( 1 - 978. iT - 1.21e7T^{2} \)
61 \( 1 + 350.T + 1.38e7T^{2} \)
67 \( 1 - 24.5T + 2.01e7T^{2} \)
71 \( 1 + 162. iT - 2.54e7T^{2} \)
73 \( 1 + 1.45e3T + 2.83e7T^{2} \)
79 \( 1 - 1.21e4T + 3.89e7T^{2} \)
83 \( 1 + 3.47e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.75e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.50e3T + 8.85e7T^{2} \)
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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

−8.968565778098013490196724403099, −8.623440357900246360143973683170, −8.066283295146628855774930282564, −6.71827786514057450677953568655, −5.97846731239746181604494400103, −5.42001223904793601971619270735, −3.98905131689427652285168661946, −3.42590062080290421043576420073, −1.55381401603853506552119747040, −0.49678324598770265075595865843, 1.00607905567469007805570307356, 2.15898030769220523766273921371, 3.08217609240222035998315557663, 4.26704447425099172775542841195, 4.92542031283164662626280074675, 6.34914362404427169200407619789, 6.98129687744226179870553365339, 8.119208009695231096224741044915, 8.968484086833964282885024384604, 9.754698087518179367250099059322

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