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Properties

Label 1-4004-4004.1523-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.722 - 0.691i$
Analytic cond. $18.5944$
Root an. cond. $18.5944$
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  + (−0.309 − 0.951i)3-s + (−0.406 − 0.913i)5-s + (−0.809 + 0.587i)9-s + (−0.743 + 0.669i)15-s + (−0.913 + 0.406i)17-s + (−0.951 + 0.309i)19-s + (−0.5 − 0.866i)23-s + (−0.669 + 0.743i)25-s + (0.809 + 0.587i)27-s + (0.669 + 0.743i)29-s + (−0.406 + 0.913i)31-s + (−0.207 − 0.978i)37-s + (−0.207 + 0.978i)41-s + (−0.5 − 0.866i)43-s + (0.866 + 0.5i)45-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.406 − 0.913i)5-s + (−0.809 + 0.587i)9-s + (−0.743 + 0.669i)15-s + (−0.913 + 0.406i)17-s + (−0.951 + 0.309i)19-s + (−0.5 − 0.866i)23-s + (−0.669 + 0.743i)25-s + (0.809 + 0.587i)27-s + (0.669 + 0.743i)29-s + (−0.406 + 0.913i)31-s + (−0.207 − 0.978i)37-s + (−0.207 + 0.978i)41-s + (−0.5 − 0.866i)43-s + (0.866 + 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7564387398 - 0.3035546072i\)
\(L(\frac12)\) \(\approx\) \(0.7564387398 - 0.3035546072i\)
\(L(1)\) \(\approx\) \(0.6866707227 - 0.2826987548i\)
\(L(1)\) \(\approx\) \(0.6866707227 - 0.2826987548i\)

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 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.406 - 0.913i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.406 + 0.913i)T \)
37 \( 1 + (-0.207 - 0.978i)T \)
41 \( 1 + (-0.207 + 0.978i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.743 - 0.669i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.994 - 0.104i)T \)
73 \( 1 + (-0.207 - 0.978i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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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

−18.479074730566652750086321369086, −17.78338457638705383788360508676, −17.263525109222460075376019774751, −16.409754948031505131711253774709, −15.76097163315158001125771029956, −15.07642703485714295986160204560, −14.888011682727611002317699122590, −13.792372061298492869896832229273, −13.316386091310149184702499628136, −12.00673287326930269408675349160, −11.62711838339995205662428222197, −10.95939142975300438731216170675, −10.31426647929121713762618210451, −9.756977679236953182352313375748, −8.88095902606038088226737819957, −8.210663694951961393459698712951, −7.27175477833911156577368060692, −6.51003962843061484068786003923, −5.93718169803345746377567443220, −4.960166754213867724182755912835, −4.21798226841634137490910051163, −3.65530003272632174983018342898, −2.77657573023641454506156516058, −2.05551479996645577607781553802, −0.40787352316209814454968580000, 0.56385795242957201847172196926, 1.58636192120176562498038719530, 2.16655975320367931413125523749, 3.27287299238109968227257748190, 4.30049863930807481989994862226, 4.8802596455108589011446494937, 5.77579610665309093120878818057, 6.46331047118623434585458354648, 7.147621874384644586579876304887, 8.008977188745753909948935159318, 8.60266754215982863046657126333, 9.013750902842744334356391261259, 10.323850311571167675586036463889, 10.88213545739526372524189592174, 11.7260896688735181769613111267, 12.369729541536854076135933593817, 12.80068368291699173070790433218, 13.39439836724908729974466578692, 14.258816572043955127103917020331, 14.90400778659611168516646228865, 15.94995637208964614984488434265, 16.36439158811003819839007335750, 17.15522309578706980765189353560, 17.65924910345959774291750588268, 18.380274830163123747700429504464

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