Properties

Degree $2$
Conductor $882$
Sign $0.939 + 0.342i$
Motivic weight $1$
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 + (−1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s − 1.73i·6-s − 0.999·8-s + (1.5 + 2.59i)9-s + 3·10-s + (1.5 + 2.59i)11-s + (1.49 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−4.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s − 3·17-s + (−1.5 + 2.59i)18-s + 7·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s − 0.707i·6-s − 0.353·8-s + (0.5 + 0.866i)9-s + 0.948·10-s + (0.452 + 0.783i)11-s + (0.433 − 0.250i)12-s + (−0.138 + 0.240i)13-s + (−1.16 + 0.670i)15-s + (−0.125 − 0.216i)16-s − 0.727·17-s + (−0.353 + 0.612i)18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.939 + 0.342i$
Motivic weight: \(1\)
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55024 - 0.273350i\)
\(L(\frac12)\) \(\approx\) \(1.55024 - 0.273350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.877447480455617376993334873529, −9.255505984517883432505675079881, −8.301183231056055855344608894092, −7.29117996637768372246160041753, −6.57704481571504673144800439881, −5.68671321128062997522276327147, −4.91636534811541873833526297319, −4.30918883849380669987224604026, −2.28667728761921153767464113895, −0.908358593200701456434204711347, 1.27136214522902862082356429554, 2.96324445485016578225124427418, 3.59513487770469630493419161079, 5.01123085682411587890281074804, 5.67476906995538082895750696529, 6.54754633643403299961289665079, 7.28306231394502376893488186712, 8.936261980336884443152273871707, 9.673463905284701070644229080629, 10.31890396833786160270954866036

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