Properties

Label 2-882-7.6-c2-0-30
Degree $2$
Conductor $882$
Sign $-0.755 + 0.654i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s − 4.89i·5-s + 2.82·8-s − 6.92i·10-s − 16.9·11-s + 1.73i·13-s + 4.00·16-s − 4.89i·17-s − 29.4i·19-s − 9.79i·20-s − 24·22-s − 8.48·23-s + 1.00·25-s + 2.44i·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s − 0.979i·5-s + 0.353·8-s − 0.692i·10-s − 1.54·11-s + 0.133i·13-s + 0.250·16-s − 0.288i·17-s − 1.54i·19-s − 0.489i·20-s − 1.09·22-s − 0.368·23-s + 0.0400·25-s + 0.0942i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.605768898\)
\(L(\frac12)\) \(\approx\) \(1.605768898\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4.89iT - 25T^{2} \)
11 \( 1 + 16.9T + 121T^{2} \)
13 \( 1 - 1.73iT - 169T^{2} \)
17 \( 1 + 4.89iT - 289T^{2} \)
19 \( 1 + 29.4iT - 361T^{2} \)
23 \( 1 + 8.48T + 529T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + 12.1iT - 961T^{2} \)
37 \( 1 + 47T + 1.36e3T^{2} \)
41 \( 1 + 68.5iT - 1.68e3T^{2} \)
43 \( 1 - 31T + 1.84e3T^{2} \)
47 \( 1 - 83.2iT - 2.20e3T^{2} \)
53 \( 1 + 76.3T + 2.80e3T^{2} \)
59 \( 1 + 83.2iT - 3.48e3T^{2} \)
61 \( 1 - 83.1iT - 3.72e3T^{2} \)
67 \( 1 + 31T + 4.48e3T^{2} \)
71 \( 1 - 59.3T + 5.04e3T^{2} \)
73 \( 1 + 81.4iT - 5.32e3T^{2} \)
79 \( 1 - 41T + 6.24e3T^{2} \)
83 \( 1 - 4.89iT - 6.88e3T^{2} \)
89 \( 1 + 58.7iT - 7.92e3T^{2} \)
97 \( 1 + 41.5iT - 9.40e3T^{2} \)
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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

−9.542465168879618220422181558178, −8.792844981555143053569687030934, −7.80667052689276929211478178385, −7.08591055762002227522796539815, −5.81677267436695032583240872283, −5.08900939356351530314438707525, −4.46050875014091990241514592414, −3.12119082862025654780059277589, −2.03953776651752591968572251027, −0.37392850919022584629115410885, 1.91609415668378748187104919670, 2.98512590766695091020773263592, 3.77923312379817799124356932150, 5.09567454881088082301706078641, 5.82935132901021588695285958400, 6.74483563311812694998883587945, 7.65996595947627949013365827103, 8.257506656419950973333116845842, 9.741494498776663041630390263397, 10.51623410734160042835571453093

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