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Properties

Label 2-3872-88.3-c0-0-1
Degree $2$
Conductor $3872$
Sign $0.970 + 0.242i$
Analytic cond. $1.93237$
Root an. cond. $1.39010$
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  + (−0.190 − 0.587i)3-s + (0.5 − 0.363i)9-s + (1.30 + 0.951i)17-s + (0.5 + 1.53i)19-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.190 + 0.587i)41-s − 0.618·43-s + (−0.809 − 0.587i)49-s + (0.309 − 0.951i)51-s + (0.809 − 0.587i)57-s + (0.5 − 1.53i)59-s + 1.61·67-s + (0.190 − 0.587i)73-s + (0.5 − 0.363i)75-s + ⋯
L(s)  = 1  + (−0.190 − 0.587i)3-s + (0.5 − 0.363i)9-s + (1.30 + 0.951i)17-s + (0.5 + 1.53i)19-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.190 + 0.587i)41-s − 0.618·43-s + (−0.809 − 0.587i)49-s + (0.309 − 0.951i)51-s + (0.809 − 0.587i)57-s + (0.5 − 1.53i)59-s + 1.61·67-s + (0.190 − 0.587i)73-s + (0.5 − 0.363i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3872\)    =    \(2^{5} \cdot 11^{2}\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3872} (1455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3872,\ (\ :0),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341064116\)
\(L(\frac12)\) \(\approx\) \(1.341064116\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 - 1.61T + T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.326050721334343529810883974680, −7.935421209579347632186535467520, −7.17524784941123476321920731668, −6.41975433485721405781453964527, −5.76508775386321506666672070296, −5.02472956278491721554359952515, −3.78971209761155360699784115861, −3.37274262229369935377825936310, −1.87176521693342426096436681134, −1.17838969455477199886323253389, 0.981682183584508824880832420051, 2.39669458012458752597583341944, 3.24805215846863542836296870444, 4.22374558718100441281771083578, 4.98562081994747938582186822852, 5.43470860799048398654561244468, 6.55322448122963253208779282278, 7.26198679935756144741850182064, 7.86118758419897716921171784890, 8.809581134972902504610123131467

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