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Properties

Label 2-882-441.142-c1-0-6
Degree $2$
Conductor $882$
Sign $-0.999 + 0.0373i$
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 + (−1.08 + 1.34i)3-s + (−0.733 − 0.680i)4-s + (−0.0755 + 0.0363i)5-s + (−0.856 − 1.50i)6-s + (1.62 − 2.08i)7-s + (0.900 − 0.433i)8-s + (−0.628 − 2.93i)9-s + (−0.00626 − 0.0835i)10-s + (0.316 + 0.397i)11-s + (1.71 − 0.246i)12-s + (−1.94 + 4.95i)13-s + (1.34 + 2.27i)14-s + (0.0332 − 0.141i)15-s + (0.0747 + 0.997i)16-s + (1.92 − 1.78i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.658i)2-s + (−0.628 + 0.777i)3-s + (−0.366 − 0.340i)4-s + (−0.0337 + 0.0162i)5-s + (−0.349 − 0.614i)6-s + (0.614 − 0.788i)7-s + (0.318 − 0.153i)8-s + (−0.209 − 0.977i)9-s + (−0.00198 − 0.0264i)10-s + (0.0955 + 0.119i)11-s + (0.494 − 0.0712i)12-s + (−0.539 + 1.37i)13-s + (0.360 + 0.608i)14-s + (0.00858 − 0.0364i)15-s + (0.0186 + 0.249i)16-s + (0.467 − 0.434i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0373i$
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: \((2,\ 882,\ (\ :1/2),\ -0.999 + 0.0373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0115280 - 0.616996i\)
\(L(\frac12)\) \(\approx\) \(0.0115280 - 0.616996i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 1.34i)T \)
7 \( 1 + (-1.62 + 2.08i)T \)
good5 \( 1 + (0.0755 - 0.0363i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.316 - 0.397i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.94 - 4.95i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-1.92 + 1.78i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.58 - 4.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.209 - 0.917i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.47 + 2.61i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (4.62 - 8.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.66 - 1.74i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-6.57 - 4.48i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (0.535 - 0.364i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.50 + 3.83i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (9.98 - 3.07i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.44 + 0.988i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-0.512 + 0.475i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (3.53 - 6.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.06 + 9.05i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (14.5 + 2.19i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-1.75 - 3.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.306 - 0.781i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-3.96 - 10.1i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.892 + 1.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.46684694100430555799625098720, −9.606522327210855065810126483733, −9.121761494526752090407785072352, −7.85991886040156137715736942112, −7.18535888500653983146042621994, −6.23379052481598275022859671783, −5.30307091687220074243847179489, −4.43472143257764794127673110421, −3.72034508036065835587026783665, −1.56438858787980412959649849449, 0.35436335887778126566422501695, 1.89635661435797136030446973483, 2.76876317315596961659023359406, 4.33896761305050011572718980029, 5.45590792823611438945818004832, 6.00189064965776126138113188805, 7.47357973824836322402266343565, 7.924589038636670490440676046295, 8.854044395071673276858580135287, 9.810800549486766599172001839748

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