Properties

Label 2-882-9.4-c1-0-1
Degree $2$
Conductor $882$
Sign $-0.999 - 0.0334i$
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 + (0.619 − 1.61i)3-s + (−0.499 + 0.866i)4-s + (−1.59 + 2.75i)5-s + (1.71 − 0.272i)6-s − 0.999·8-s + (−2.23 − 2.00i)9-s − 3.18·10-s + (−1.59 − 2.75i)11-s + (1.09 + 1.34i)12-s + (−2.85 + 4.93i)13-s + (3.47 + 4.28i)15-s + (−0.5 − 0.866i)16-s − 1.52·17-s + (0.619 − 2.93i)18-s − 1.28·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.357 − 0.933i)3-s + (−0.249 + 0.433i)4-s + (−0.711 + 1.23i)5-s + (0.698 − 0.111i)6-s − 0.353·8-s + (−0.744 − 0.668i)9-s − 1.00·10-s + (−0.479 − 0.830i)11-s + (0.314 + 0.388i)12-s + (−0.790 + 1.36i)13-s + (0.896 + 1.10i)15-s + (−0.125 − 0.216i)16-s − 0.369·17-s + (0.146 − 0.691i)18-s − 0.294·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 - 0.0334i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.999 - 0.0334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00882256 + 0.526913i\)
\(L(\frac12)\) \(\approx\) \(0.00882256 + 0.526913i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 + 1.61i)T \)
7 \( 1 \)
good5 \( 1 + (1.59 - 2.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.85 - 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.52T + 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.71 - 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 + (0.562 - 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.03 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.225T + 89T^{2} \)
97 \( 1 + (7.42 + 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.80303558538234928949019964571, −9.474691883173347633635106713020, −8.582347397632892340857827451008, −7.72892013012262049871044440148, −7.09760411221788208643661980369, −6.56185567469204515968154738936, −5.58130357992968394428316705897, −4.13041015119845681670262599970, −3.20909594041700943722037861058, −2.21361231639694213251103570002, 0.19878543287695356164102709028, 2.18468639422949175003931924685, 3.37024481652465288313903189158, 4.40000979252967198994723719982, 4.91478578041361297712965995499, 5.70380712307994137359587174269, 7.52918457935349678126897742996, 8.166564844498692937028640666992, 9.093887721988965283285654096177, 9.720676249980402266599786939067

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