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Properties

Label 2-3864-3864.587-c0-0-3
Degree $2$
Conductor $3864$
Sign $-0.451 + 0.892i$
Analytic cond. $1.92838$
Root an. cond. $1.38866$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.841 + 0.540i)12-s + (−1.10 + 1.27i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.797 − 1.74i)17-s + (−0.959 + 0.281i)18-s + (0.841 − 0.540i)21-s + (−0.142 − 0.989i)23-s + 0.999·24-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.841 + 0.540i)12-s + (−1.10 + 1.27i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.797 − 1.74i)17-s + (−0.959 + 0.281i)18-s + (0.841 − 0.540i)21-s + (−0.142 − 0.989i)23-s + 0.999·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.213613838\)
\(L(\frac12)\) \(\approx\) \(1.213613838\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
good5 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + (-0.841 - 0.540i)T^{2} \)
41 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
67 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−8.825185022159217751516729771206, −7.31978235594298868200546950697, −6.84557468930223431139491559771, −6.07366028065201316632967760144, −5.11085976580862478720533109793, −4.30892492269684042735122245380, −4.20858776000602135996290314754, −2.86483347037194287495241753544, −2.41008503113107726781268003730, −0.48016662795389307740467407595, 1.74640187898152114819406134363, 2.76292419439472452721908713278, 3.27487657624048786157729286895, 4.52502523982885136453370669032, 5.52054434001642224922145606792, 5.84700860419203634464246002447, 6.60873054558669844047380725013, 7.32008475655946707656761040420, 7.940186365311908373897241144125, 8.613414470376950088975795405096

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