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Properties

Label 2-3864-3864.1469-c0-0-4
Degree $2$
Conductor $3864$
Sign $0.716 + 0.697i$
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  + (−0.989 + 0.142i)2-s + (−0.997 − 0.0713i)3-s + (0.959 − 0.281i)4-s + (1.27 − 1.47i)5-s + (0.997 − 0.0713i)6-s + (0.540 + 0.841i)7-s + (−0.909 + 0.415i)8-s + (0.989 + 0.142i)9-s + (−1.05 + 1.64i)10-s + (−0.977 + 0.212i)12-s + (1.00 + 0.647i)13-s + (−0.654 − 0.755i)14-s + (−1.38 + 1.38i)15-s + (0.841 − 0.540i)16-s − 0.999·18-s + (−0.562 − 1.91i)19-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (−0.997 − 0.0713i)3-s + (0.959 − 0.281i)4-s + (1.27 − 1.47i)5-s + (0.997 − 0.0713i)6-s + (0.540 + 0.841i)7-s + (−0.909 + 0.415i)8-s + (0.989 + 0.142i)9-s + (−1.05 + 1.64i)10-s + (−0.977 + 0.212i)12-s + (1.00 + 0.647i)13-s + (−0.654 − 0.755i)14-s + (−1.38 + 1.38i)15-s + (0.841 − 0.540i)16-s − 0.999·18-s + (−0.562 − 1.91i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ 0.716 + 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8858662625\)
\(L(\frac12)\) \(\approx\) \(0.8858662625\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 - 0.142i)T \)
3 \( 1 + (0.997 + 0.0713i)T \)
7 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good5 \( 1 + (-1.27 + 1.47i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-1.00 - 0.647i)T + (0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (0.562 + 1.91i)T + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (-0.142 - 0.989i)T^{2} \)
43 \( 1 + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.0771 + 0.120i)T + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (-1.28 + 0.587i)T + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.281 - 0.0405i)T + (0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.627 - 0.724i)T + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (-0.142 - 0.989i)T^{2} \)
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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

−8.740693180710303303124572279568, −8.139945207046180747618340189491, −6.87485776372137369604475377542, −6.40952371515992502378718256220, −5.53609891418283605772108713526, −5.22195496226775841772066595040, −4.36910732143697834785379890263, −2.48306103128443245666052326763, −1.67194955311729357708698991887, −0.941800410524192995628633617446, 1.23424377950804259953774246217, 1.95720822107119571971228713709, 3.16880259800652640254563341358, 3.96241113353308000439001982597, 5.41908216291901269537679730542, 6.05552591675020186021802574629, 6.52831246228277804012206598305, 7.22134863893637385799186929630, 7.87207535392873357818840004744, 8.826043929561271538572663672079

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