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Properties

Label 2-882-63.20-c1-0-37
Degree $2$
Conductor $882$
Sign $-0.541 - 0.840i$
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 + (0.840 − 1.51i)3-s + (0.499 + 0.866i)4-s + (−1.14 − 1.97i)5-s + (−1.48 + 0.890i)6-s − 0.999i·8-s + (−1.58 − 2.54i)9-s + 2.28i·10-s + (0.946 + 0.546i)11-s + (1.73 − 0.0288i)12-s + (−5.91 + 3.41i)13-s + (−3.95 + 0.0659i)15-s + (−0.5 + 0.866i)16-s − 6.71·17-s + (0.100 + 2.99i)18-s + 2.86i·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.485 − 0.874i)3-s + (0.249 + 0.433i)4-s + (−0.510 − 0.883i)5-s + (−0.606 + 0.363i)6-s − 0.353i·8-s + (−0.528 − 0.848i)9-s + 0.721i·10-s + (0.285 + 0.164i)11-s + (0.499 − 0.00833i)12-s + (−1.64 + 0.947i)13-s + (−1.02 + 0.0170i)15-s + (−0.125 + 0.216i)16-s − 1.62·17-s + (0.0235 + 0.706i)18-s + 0.656i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.541 - 0.840i$
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: \((2,\ 882,\ (\ :1/2),\ -0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133648 + 0.245150i\)
\(L(\frac12)\) \(\approx\) \(0.133648 + 0.245150i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.840 + 1.51i)T \)
7 \( 1 \)
good5 \( 1 + (1.14 + 1.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.946 - 0.546i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.91 - 3.41i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.71T + 17T^{2} \)
19 \( 1 - 2.86iT - 19T^{2} \)
23 \( 1 + (-3.38 + 1.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.59 - 0.923i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.75 + 1.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.15T + 37T^{2} \)
41 \( 1 + (2.45 + 4.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.74 - 6.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.40 + 5.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.256iT - 53T^{2} \)
59 \( 1 + (-0.971 - 1.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.15 + 0.665i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.233iT - 71T^{2} \)
73 \( 1 - 6.80iT - 73T^{2} \)
79 \( 1 + (-3.63 + 6.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.91 + 5.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.9T + 89T^{2} \)
97 \( 1 + (-4.13 - 2.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.241302835741551991414842855411, −8.821078333816573405177922383856, −8.057824066414067971797691373125, −7.10013354275110958833298409326, −6.60294184283833517227098768482, −4.94685382976240457570866855140, −4.05245621417163986564831583246, −2.61015450832487647447106124394, −1.67227103743167397777366654968, −0.14450537563267645837624514832, 2.42321148796471604156262472023, 3.20672893417548505597648043644, 4.52846018266756996967629463848, 5.33766169099911222646180482486, 6.75976740953751911663446975196, 7.31287480378699861326795270528, 8.257243800522305974200532074160, 9.041567948508120191850455083910, 9.793430093534675029083363816847, 10.61091306832712292988541090626

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