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Properties

Label 2-882-63.20-c1-0-29
Degree $2$
Conductor $882$
Sign $0.147 - 0.989i$
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 + (1.71 + 0.264i)3-s + (0.499 + 0.866i)4-s + (1.77 + 3.07i)5-s + (1.35 + 1.08i)6-s + 0.999i·8-s + (2.85 + 0.906i)9-s + 3.55i·10-s + (−2.61 − 1.51i)11-s + (0.626 + 1.61i)12-s + (−0.888 + 0.513i)13-s + (2.22 + 5.73i)15-s + (−0.5 + 0.866i)16-s + 1.61·17-s + (2.02 + 2.21i)18-s − 8.22i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.988 + 0.152i)3-s + (0.249 + 0.433i)4-s + (0.794 + 1.37i)5-s + (0.551 + 0.442i)6-s + 0.353i·8-s + (0.953 + 0.302i)9-s + 1.12i·10-s + (−0.789 − 0.455i)11-s + (0.180 + 0.466i)12-s + (−0.246 + 0.142i)13-s + (0.574 + 1.48i)15-s + (−0.125 + 0.216i)16-s + 0.392·17-s + (0.476 + 0.522i)18-s − 1.88i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.147 - 0.989i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56932 + 2.21485i\)
\(L(\frac12)\) \(\approx\) \(2.56932 + 2.21485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.264i)T \)
7 \( 1 \)
good5 \( 1 + (-1.77 - 3.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.61 + 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.888 - 0.513i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + 8.22iT - 19T^{2} \)
23 \( 1 + (2.90 - 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.70 + 2.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.18 + 2.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + (-0.0472 - 0.0817i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.57 + 4.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.18iT - 53T^{2} \)
59 \( 1 + (-4.42 - 7.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.06 + 2.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.187 - 0.325i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + 1.31iT - 73T^{2} \)
79 \( 1 + (0.462 - 0.800i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.43 + 9.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.70T + 89T^{2} \)
97 \( 1 + (-13.3 - 7.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31773670558876871144786385826, −9.529283269116551960985485763904, −8.589699031168065934454568070552, −7.51606981036690370624126612465, −7.02479175206100706799205836595, −6.04317589266762235838329268686, −5.06486690205397779167489037593, −3.81526367271740658324772454686, −2.80768051523945549033807301908, −2.29647793179910922339432707762, 1.38954440153447791948924196278, 2.22096826051998805853263170167, 3.50421263962333652713603767567, 4.56959320795389734518506662794, 5.34571046014919679774916131859, 6.27151988320243374154414415192, 7.64868620487261135010927397254, 8.265114080787718883099877206086, 9.210146063877125594572872571753, 9.958178518452828446617928797956

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