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Properties

Label 2-3864-3864.587-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5374756052\)
\(L(\frac12)\) \(\approx\) \(0.5374756052\)
\(L(1)\) 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

−8.732219061089320407403785601128, −7.86169464852417514032246991558, −7.38025035913659136629889085494, −6.72267853886752990070102315835, −6.22715010716141004624461545635, −5.38096436913293597168342166273, −4.24909205718288533012059452082, −3.12578068713405995343580641640, −2.00985949013689289383204081199, −1.19831140607411464383741161286, 0.41785987883937374187561179182, 2.32754005477845840637814540909, 3.05056123472859725104801230654, 3.40526470826717973680260644981, 4.95401250485053589920160398342, 5.18959450856493468059972819419, 6.45503092680635714471558278523, 7.25219392572824899220396592689, 8.084105895549959306411170692992, 8.709992127889784326548367868008

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