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Properties

Label 2-882-441.142-c1-0-28
Degree $2$
Conductor $882$
Sign $-0.139 + 0.990i$
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  + (0.365 − 0.930i)2-s + (−1.60 − 0.655i)3-s + (−0.733 − 0.680i)4-s + (−0.804 + 0.387i)5-s + (−1.19 + 1.25i)6-s + (−0.673 + 2.55i)7-s + (−0.900 + 0.433i)8-s + (2.14 + 2.10i)9-s + (0.0667 + 0.890i)10-s + (0.0570 + 0.0715i)11-s + (0.729 + 1.57i)12-s + (0.0452 − 0.115i)13-s + (2.13 + 1.56i)14-s + (1.54 − 0.0942i)15-s + (0.0747 + 0.997i)16-s + (4.72 − 4.38i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.658i)2-s + (−0.925 − 0.378i)3-s + (−0.366 − 0.340i)4-s + (−0.359 + 0.173i)5-s + (−0.488 + 0.511i)6-s + (−0.254 + 0.967i)7-s + (−0.318 + 0.153i)8-s + (0.713 + 0.700i)9-s + (0.0210 + 0.281i)10-s + (0.0171 + 0.0215i)11-s + (0.210 + 0.453i)12-s + (0.0125 − 0.0319i)13-s + (0.570 + 0.417i)14-s + (0.398 − 0.0243i)15-s + (0.0186 + 0.249i)16-s + (1.14 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681218 - 0.783637i\)
\(L(\frac12)\) \(\approx\) \(0.681218 - 0.783637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
3 \( 1 + (1.60 + 0.655i)T \)
7 \( 1 + (0.673 - 2.55i)T \)
good5 \( 1 + (0.804 - 0.387i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.0570 - 0.0715i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.0452 + 0.115i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-4.72 + 4.38i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.76 + 4.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.969 + 4.24i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (3.69 + 1.13i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-2.68 + 4.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.24 - 0.384i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-6.90 - 4.70i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (7.49 - 5.10i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (0.0117 - 0.0298i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-6.08 + 1.87i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-9.26 + 6.31i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-4.97 + 4.61i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-4.61 + 7.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.48 + 6.51i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.76 - 1.47i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (1.46 + 2.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.37 + 13.6i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.870 - 2.21i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.0541 + 0.0937i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.872739847164244375428363989682, −9.458094565499971295072044087474, −8.164531617844568197309639079726, −7.26187888055973325610430770349, −6.27732723958008321374278054559, −5.39864267250832933975912932861, −4.72709372472569501448212179324, −3.31536363712250150952789082361, −2.26842009452350477223769142301, −0.65404579798520764372946139584, 1.09981468749715053178354642919, 3.67702761867699526373627475754, 3.98092720271061127714302885495, 5.30536884501446301517253729218, 5.87087752084990748453235537920, 6.93537600429470950389141956896, 7.59764927105623647721349033778, 8.504316334254884191046180519923, 9.842686293188259049875955115214, 10.16796598167250512992629726803

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