Properties

Label 2-882-7.6-c4-0-15
Degree $2$
Conductor $882$
Sign $0.755 - 0.654i$
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  − 2.82·2-s + 8.00·4-s − 14.1i·5-s − 22.6·8-s + 40.0i·10-s − 64.0·11-s − 228. i·13-s + 64.0·16-s + 225. i·17-s + 294. i·19-s − 113. i·20-s + 181.·22-s − 709.·23-s + 424.·25-s + 647. i·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s − 0.566i·5-s − 0.353·8-s + 0.400i·10-s − 0.529·11-s − 1.35i·13-s + 0.250·16-s + 0.780i·17-s + 0.816i·19-s − 0.283i·20-s + 0.374·22-s − 1.34·23-s + 0.679·25-s + 0.957i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(91.1723\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :2),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9346286134\)
\(L(\frac12)\) \(\approx\) \(0.9346286134\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 14.1iT - 625T^{2} \)
11 \( 1 + 64.0T + 1.46e4T^{2} \)
13 \( 1 + 228. iT - 2.85e4T^{2} \)
17 \( 1 - 225. iT - 8.35e4T^{2} \)
19 \( 1 - 294. iT - 1.30e5T^{2} \)
23 \( 1 + 709.T + 2.79e5T^{2} \)
29 \( 1 + 740.T + 7.07e5T^{2} \)
31 \( 1 + 666. iT - 9.23e5T^{2} \)
37 \( 1 + 833.T + 1.87e6T^{2} \)
41 \( 1 - 2.81e3iT - 2.82e6T^{2} \)
43 \( 1 - 3.06e3T + 3.41e6T^{2} \)
47 \( 1 + 613. iT - 4.87e6T^{2} \)
53 \( 1 - 1.15e3T + 7.89e6T^{2} \)
59 \( 1 - 3.49e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.27e3iT - 1.38e7T^{2} \)
67 \( 1 + 8.67e3T + 2.01e7T^{2} \)
71 \( 1 - 353.T + 2.54e7T^{2} \)
73 \( 1 - 4.06e3iT - 2.83e7T^{2} \)
79 \( 1 - 6.47e3T + 3.89e7T^{2} \)
83 \( 1 - 8.22e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.55e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.55e3iT - 8.85e7T^{2} \)
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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

−9.747881467029459488976783179596, −8.697168891970709683346484187572, −8.041016023333142060884345155488, −7.48046269782203192123591810374, −6.07400520878187717453397285635, −5.54961620649477422415770001354, −4.25833522281845272503458479122, −3.10031177059807273507657361526, −1.89239657980900857657407972131, −0.75087166396158790246634180069, 0.35693736436136417517232119515, 1.86551390396101856228195060760, 2.72901191265182802405390759427, 3.95851010901310512558202141697, 5.13606954745598223907190740738, 6.26761334886667028840214639156, 7.08437843072367460745809904548, 7.64488407013448962942997102215, 8.886419915999651717181002906420, 9.294239966246246008805166952868

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