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Properties

Label 2-882-63.59-c1-0-22
Degree $2$
Conductor $882$
Sign $0.996 + 0.0825i$
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  + (0.866 + 0.5i)2-s + (−1.57 + 0.716i)3-s + (0.499 + 0.866i)4-s − 2.34·5-s + (−1.72 − 0.167i)6-s + 0.999i·8-s + (1.97 − 2.26i)9-s + (−2.03 − 1.17i)10-s − 5.67i·11-s + (−1.40 − 1.00i)12-s + (1.48 + 0.859i)13-s + (3.70 − 1.68i)15-s + (−0.5 + 0.866i)16-s + (0.884 − 1.53i)17-s + (2.83 − 0.971i)18-s + (0.986 − 0.569i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.910 + 0.413i)3-s + (0.249 + 0.433i)4-s − 1.05·5-s + (−0.703 − 0.0684i)6-s + 0.353i·8-s + (0.657 − 0.753i)9-s + (−0.643 − 0.371i)10-s − 1.71i·11-s + (−0.406 − 0.290i)12-s + (0.413 + 0.238i)13-s + (0.956 − 0.434i)15-s + (−0.125 + 0.216i)16-s + (0.214 − 0.371i)17-s + (0.669 − 0.228i)18-s + (0.226 − 0.130i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.996 + 0.0825i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.996 + 0.0825i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26761 - 0.0524172i\)
\(L(\frac12)\) \(\approx\) \(1.26761 - 0.0524172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (1.57 - 0.716i)T \)
7 \( 1 \)
good5 \( 1 + 2.34T + 5T^{2} \)
11 \( 1 + 5.67iT - 11T^{2} \)
13 \( 1 + (-1.48 - 0.859i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.884 + 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.986 + 0.569i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.67iT - 23T^{2} \)
29 \( 1 + (-3.59 + 2.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.24 + 4.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.99 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 3.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.90 + 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.11 - 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.79 + 4.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.43 + 9.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.52iT - 71T^{2} \)
73 \( 1 + (-4.62 - 2.67i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.51 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.27 + 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.580 - 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.97 - 2.29i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39966609407870547820987277716, −9.231575634093412402478769890471, −8.254685715071194468527043472158, −7.51012120049874624149759015541, −6.38043048522981518318018303519, −5.82635395057567841496501087624, −4.77987130065042109839781390939, −3.91110031478756973223139454007, −3.13990323703036910311037522017, −0.68808833248343661893904699838, 1.16360112053737812405560371471, 2.60646483668053785510554646226, 4.17178961954680737724869032111, 4.57923901018414715756064347527, 5.72052424257008323861460234394, 6.68176752565436195665917963181, 7.42317968138463185843922047494, 8.178340251281466835592998431913, 9.615838804892331008762721007541, 10.46419964268729847693439922858

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