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Properties

Label 2-3864-3864.3443-c0-0-0
Degree $2$
Conductor $3864$
Sign $0.763 - 0.645i$
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  + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.654 − 0.755i)12-s + (1.25 + 0.368i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)16-s + (−0.118 + 0.822i)17-s + (0.415 + 0.909i)18-s + (−0.654 + 0.755i)21-s + (0.841 + 0.540i)23-s + 24-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.654 − 0.755i)12-s + (1.25 + 0.368i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)16-s + (−0.118 + 0.822i)17-s + (0.415 + 0.909i)18-s + (−0.654 + 0.755i)21-s + (0.841 + 0.540i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.763 - 0.645i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (3443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ 0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166870448\)
\(L(\frac12)\) \(\approx\) \(1.166870448\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good5 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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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

−8.780515960511945678285699175856, −8.059186058313606898863081792444, −7.27513853496957434998154707885, −6.67867217440671968648409962066, −6.08634690526325093903330970313, −5.33077921406521714067725840050, −3.95100136574238885158088138546, −3.33697869018018956517433476375, −2.07612083188538107356923814531, −1.17121419165583628335810495810, 0.892649465120358689758905056567, 2.39322584564261159388451992545, 2.89933048136232601366114625583, 3.91922342440500505345338891708, 4.20138003869549299630201865426, 5.54394587710170516682772028529, 6.59049748070065960547160150768, 7.38819174562183265068351825692, 8.090435902143467466234521253296, 8.754725260697683215625400827515

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