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Properties

Label 2-3888-4.3-c0-0-1
Degree $2$
Conductor $3888$
Sign $-i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
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  + 1.73i·7-s + 13-s + 1.73i·19-s − 25-s − 1.73i·31-s − 37-s + 1.73i·43-s − 1.99·49-s + 2·61-s − 2·73-s + 1.73i·79-s + 1.73i·91-s + 97-s + 109-s + ⋯
L(s)  = 1  + 1.73i·7-s + 13-s + 1.73i·19-s − 25-s − 1.73i·31-s − 37-s + 1.73i·43-s − 1.99·49-s + 2·61-s − 2·73-s + 1.73i·79-s + 1.73i·91-s + 97-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3888\)    =    \(2^{4} \cdot 3^{5}\)
Sign: $-i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3888} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3888,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198101897\)
\(L(\frac12)\) \(\approx\) \(1.198101897\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.636517345976155207467158642626, −8.304926904238248060367005804152, −7.54499874478131518908465520673, −6.29498026651160392686968601612, −5.91359528529729946373567525132, −5.36022280479461199755930395602, −4.18258059645334671716392452603, −3.40991304823595883101538525522, −2.38567725719458076953098623207, −1.59077800161339049598734438441, 0.69193233292483628424733545529, 1.79174088305807094519796510177, 3.18994212721028625485395441682, 3.83057584770096436586811897867, 4.59501332111968682976600513949, 5.40299752129183607168537666346, 6.49405015674366153155338055015, 7.04316006665404028967607017978, 7.54849808375161551318656304955, 8.591184011852339645215313924463

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