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Properties

Label 2-882-7.4-c1-0-5
Degree $2$
Conductor $882$
Sign $-0.749 - 0.661i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.41 + 2.44i)5-s − 0.999·8-s + (−1.41 + 2.44i)10-s + (−1 + 1.73i)11-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + (3.53 + 6.12i)19-s − 2.82·20-s − 1.99·22-s + (−2 − 3.46i)23-s + (−1.49 + 2.59i)25-s − 2·29-s + (−4.24 + 7.34i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.632 + 1.09i)5-s − 0.353·8-s + (−0.447 + 0.774i)10-s + (−0.301 + 0.522i)11-s + (−0.125 − 0.216i)16-s + (−0.171 + 0.297i)17-s + (0.811 + 1.40i)19-s − 0.632·20-s − 0.426·22-s + (−0.417 − 0.722i)23-s + (−0.299 + 0.519i)25-s − 0.371·29-s + (−0.762 + 1.31i)31-s + (0.0883 − 0.153i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636553 + 1.68354i\)
\(L(\frac12)\) \(\approx\) \(0.636553 + 1.68354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.53 - 6.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (4.24 - 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.41 + 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−10.41425869506081899162378999824, −9.701681255578658093529389492342, −8.706778585118666904354399286991, −7.61318556354952745724131955297, −7.05924573167588351249902334772, −6.07583927514816803550481876316, −5.48946469700608741557806251110, −4.19386852039087308500281209603, −3.16159231578217448431540122192, −2.00881355078638821036491364042, 0.77223306029558540932517534850, 2.07450558937990508322260981923, 3.27613870875071603727182958902, 4.52795059817113101429025942372, 5.29064751709623561432405187482, 5.97372687845616218937895171056, 7.27611633781654403058560036113, 8.345543149094778577021337019066, 9.351313626414997572898283439751, 9.536487193519370266617555397553

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