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Properties

Label 1-4004-4004.227-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.0860 - 0.996i$
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  + (0.913 − 0.406i)3-s + (−0.743 + 0.669i)5-s + (0.669 − 0.743i)9-s + (−0.406 + 0.913i)15-s + (−0.309 + 0.951i)17-s + (−0.994 − 0.104i)19-s + 23-s + (0.104 − 0.994i)25-s + (0.309 − 0.951i)27-s + (−0.913 − 0.406i)29-s + (−0.743 − 0.669i)31-s + (−0.587 − 0.809i)37-s + (0.994 + 0.104i)41-s + (0.5 − 0.866i)43-s + i·45-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.743 + 0.669i)5-s + (0.669 − 0.743i)9-s + (−0.406 + 0.913i)15-s + (−0.309 + 0.951i)17-s + (−0.994 − 0.104i)19-s + 23-s + (0.104 − 0.994i)25-s + (0.309 − 0.951i)27-s + (−0.913 − 0.406i)29-s + (−0.743 − 0.669i)31-s + (−0.587 − 0.809i)37-s + (0.994 + 0.104i)41-s + (0.5 − 0.866i)43-s + i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0860 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0860 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.0860 - 0.996i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.0860 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9535405219 - 1.039406554i\)
\(L(\frac12)\) \(\approx\) \(0.9535405219 - 1.039406554i\)
\(L(1)\) \(\approx\) \(1.130263298 - 0.1709460905i\)
\(L(1)\) \(\approx\) \(1.130263298 - 0.1709460905i\)

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

−18.84643736734812660302708667556, −18.1460526156954401435411878634, −17.072548785420802789360936282752, −16.46251041410051137410627470483, −15.96351618822771386871104077550, −15.08622369163768046134337978211, −14.87122364228754995346599552120, −13.85546390443057982988319990486, −13.217150743797158347976109775662, −12.62821980080689792505291891655, −11.85525824499719185140909899332, −10.94785173313816871859136856287, −10.467554922996620509103543548461, −9.25185151014567602893783788878, −9.050018288166211864355426825506, −8.36747240973029282210568384435, −7.439583713340096565673361651577, −7.13768017198350520659285459047, −5.84374285457909321431771119573, −4.83702531642876932493443189120, −4.489376556083924832491780071601, −3.58097221876712457724451504462, −2.95414073461895374214607115586, −1.99221029663822749854573565627, −1.06128431363072647581029096944, 0.359505871491571383267650730664, 1.684960733870982453180152634340, 2.37131388843978373636076601741, 3.1719890791859866944768708655, 3.94673454025745772379772262555, 4.38815526008018699440638856784, 5.788279102053623342452641764668, 6.47594182025370695982929822712, 7.359078511132342586416288750743, 7.6167920760020323212511668177, 8.6437735742117157841581579027, 8.98829477388173008657131831193, 10.00871313090666991903370212876, 10.82675690184178993533704097354, 11.280831442853937243907415665064, 12.383695807751776210320392952662, 12.792976944075473537880626222872, 13.51506949323219777170537551518, 14.42141215726546034329109901399, 14.86916721151217444406349967285, 15.3417126868690091687477507215, 16.05847471481654519609519619645, 17.0805832867639087269594745750, 17.67887574482249737498416405325, 18.63971210537959281499519727631

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