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Properties

Label 2-882-441.142-c1-0-29
Degree $2$
Conductor $882$
Sign $0.108 - 0.994i$
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 + 0.651i)3-s + (−0.733 − 0.680i)4-s + (3.48 − 1.67i)5-s + (−1.19 + 1.25i)6-s + (−0.638 + 2.56i)7-s + (0.900 − 0.433i)8-s + (2.15 + 2.09i)9-s + (0.289 + 3.85i)10-s + (−0.807 − 1.01i)11-s + (−0.733 − 1.56i)12-s + (−1.91 + 4.87i)13-s + (−2.15 − 1.53i)14-s + (6.68 − 0.422i)15-s + (0.0747 + 0.997i)16-s + (−3.80 + 3.53i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.658i)2-s + (0.926 + 0.376i)3-s + (−0.366 − 0.340i)4-s + (1.55 − 0.750i)5-s + (−0.486 + 0.512i)6-s + (−0.241 + 0.970i)7-s + (0.318 − 0.153i)8-s + (0.716 + 0.697i)9-s + (0.0913 + 1.21i)10-s + (−0.243 − 0.305i)11-s + (−0.211 − 0.452i)12-s + (−0.530 + 1.35i)13-s + (−0.576 − 0.409i)14-s + (1.72 − 0.109i)15-s + (0.0186 + 0.249i)16-s + (−0.923 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.108 - 0.994i$
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: \((2,\ 882,\ (\ :1/2),\ 0.108 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66445 + 1.49293i\)
\(L(\frac12)\) \(\approx\) \(1.66445 + 1.49293i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 - 0.651i)T \)
7 \( 1 + (0.638 - 2.56i)T \)
good5 \( 1 + (-3.48 + 1.67i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (0.807 + 1.01i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.91 - 4.87i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (3.80 - 3.53i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (1.33 - 2.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.31 + 5.76i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-6.91 - 2.13i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-1.21 + 2.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.131 + 0.0405i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-4.08 - 2.78i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-5.37 + 3.66i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.44 + 3.68i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (4.90 - 1.51i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-8.38 + 5.71i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-6.61 + 6.13i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-3.14 + 5.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.44 + 6.32i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (14.8 + 2.23i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (7.23 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.78 + 12.1i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.493 - 1.25i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (5.41 - 9.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.961599734709629678568251792989, −9.213677061250196293448085392122, −8.812659647279808312660110587058, −8.214387884799304470055731900273, −6.69997926658770148664536331049, −6.12617909105561970882718838392, −5.06154415971826933674902767389, −4.33551678238139709388006681117, −2.53230474061147808082217698542, −1.81959494351751100381992613694, 1.14076506316715814180734551531, 2.59608080224875378162226322701, 2.84444224791088964555378268093, 4.31058273939419128383560564788, 5.57949101344817198598074342166, 6.86675818729435040700160376785, 7.27669310993868976392198996298, 8.391325828487303414599626846109, 9.449822450024441825367821273601, 9.900779004141998549242632103166

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