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Properties

Label 2-882-63.4-c1-0-20
Degree $2$
Conductor $882$
Sign $0.975 + 0.220i$
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 + 1.73i·3-s + (−0.499 − 0.866i)4-s + 2·5-s + (1.49 + 0.866i)6-s − 0.999·8-s − 2.99·9-s + (1 − 1.73i)10-s + 11-s + (1.49 − 0.866i)12-s + (3 − 5.19i)13-s + 3.46i·15-s + (−0.5 + 0.866i)16-s + (2.5 − 4.33i)17-s + (−1.49 + 2.59i)18-s + (3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + 0.999i·3-s + (−0.249 − 0.433i)4-s + 0.894·5-s + (0.612 + 0.353i)6-s − 0.353·8-s − 0.999·9-s + (0.316 − 0.547i)10-s + 0.301·11-s + (0.433 − 0.249i)12-s + (0.832 − 1.44i)13-s + 0.894i·15-s + (−0.125 + 0.216i)16-s + (0.606 − 1.05i)17-s + (−0.353 + 0.612i)18-s + (0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20166 - 0.245744i\)
\(L(\frac12)\) \(\approx\) \(2.20166 - 0.245744i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.22320619755545118655195289616, −9.532557728911292126627635924558, −8.775126213065446762474526089106, −7.73441976365842372781486087829, −6.18336546061230623342307495037, −5.53104872938569287196583548776, −4.85149636420972615815433211238, −3.46695233978356895747322902688, −2.95345304042477107874750155634, −1.24216478288915097802112651595, 1.32099734781954153468787152585, 2.51905966752056633678736527264, 3.86494639311876150628217678616, 5.15165993315668965696796025589, 6.08476828479604175178802693467, 6.58841792152257971985007080180, 7.38054044225898634615941577805, 8.465615571691745927792215072143, 9.062811038441739238282432200803, 9.976447831725577390742809063318

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