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Properties

Label 1-4004-4004.999-r1-0-0
Degree $1$
Conductor $4004$
Sign $-0.0623 + 0.998i$
Analytic cond. $430.289$
Root an. cond. $430.289$
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  + (−0.978 + 0.207i)3-s + (0.406 − 0.913i)5-s + (0.913 − 0.406i)9-s + (−0.207 + 0.978i)15-s + (−0.809 + 0.587i)17-s + (−0.743 + 0.669i)19-s + 23-s + (−0.669 − 0.743i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (0.406 + 0.913i)31-s + (0.951 − 0.309i)37-s + (0.743 − 0.669i)41-s + (−0.5 − 0.866i)43-s − i·45-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)3-s + (0.406 − 0.913i)5-s + (0.913 − 0.406i)9-s + (−0.207 + 0.978i)15-s + (−0.809 + 0.587i)17-s + (−0.743 + 0.669i)19-s + 23-s + (−0.669 − 0.743i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (0.406 + 0.913i)31-s + (0.951 − 0.309i)37-s + (0.743 − 0.669i)41-s + (−0.5 − 0.866i)43-s − i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0623 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0623 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.0623 + 0.998i$
Analytic conductor: \(430.289\)
Root analytic conductor: \(430.289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (1:\ ),\ -0.0623 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5091048878 + 0.5418760096i\)
\(L(\frac12)\) \(\approx\) \(0.5091048878 + 0.5418760096i\)
\(L(1)\) \(\approx\) \(0.7753330689 - 0.04290212884i\)
\(L(1)\) \(\approx\) \(0.7753330689 - 0.04290212884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.743 + 0.669i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.406 + 0.913i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.743 - 0.669i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.406 - 0.913i)T \)
73 \( 1 + (0.207 - 0.978i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.994 + 0.104i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.29638047244448891515167903222, −17.39970531043567995536012870772, −16.97726247828645554728322411561, −16.22759352237617846992943960881, −15.229544282959201226253185593660, −15.00937298872353852968249543423, −13.9269220074970892776301009138, −13.140799726691100042174500875071, −12.90086681516903164141212539384, −11.63251845585412179581463523042, −11.25158631745163523681531718795, −10.781844579610829940936675897082, −9.8742461242693097777507755909, −9.3608633427504575722220058185, −8.289300859974807504784081107394, −7.270905331063524549641945847590, −6.876617635935733224515955267333, −6.20002321326545387163700821559, −5.50787280720954185982770148640, −4.68272214060698648337645974804, −3.95237184309992358165505926417, −2.75480970954438492567210941343, −2.20600788174294014948813565435, −1.13105923195221603090041515558, −0.170532897396469116704902899061, 0.73416613939669851844373299261, 1.539458538375443646479047511209, 2.34745237275453529109592121388, 3.751063795962798923246855687158, 4.336244638511236688643326154914, 5.06373532187981870998718065228, 5.745999316973547487519313954242, 6.33181834499016456947473083748, 7.13439293198276766575123985510, 8.07682936992351374992080246186, 8.95528492718182483808140492930, 9.39314768078382107454567942016, 10.451148852021071407777921797937, 10.72641902322289802164399959193, 11.73725919063000283710447505275, 12.271712337820544075750902552473, 13.104496888119081177837704939386, 13.27857132308702478111338625580, 14.56512696886368287434964046822, 15.15884198793351864392900360356, 16.03631695741141705667839435989, 16.46777374934298776331089424050, 17.272663511441116766715162487303, 17.47904847186680122508500788155, 18.34102143242223455920800806644

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