↑

Properties

Label 1-4004-4004.1543-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.968 - 0.250i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.104 − 0.994i)3-s + (0.978 + 0.207i)5-s + (−0.978 + 0.207i)9-s + (0.104 − 0.994i)15-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s − 23-s + (0.913 + 0.406i)25-s + (0.309 + 0.951i)27-s + (−0.104 + 0.994i)29-s + (−0.978 + 0.207i)31-s + (−0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (0.5 − 0.866i)43-s − 45-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)3-s + (0.978 + 0.207i)5-s + (−0.978 + 0.207i)9-s + (0.104 − 0.994i)15-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s − 23-s + (0.913 + 0.406i)25-s + (0.309 + 0.951i)27-s + (−0.104 + 0.994i)29-s + (−0.978 + 0.207i)31-s + (−0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (0.5 − 0.866i)43-s − 45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1232281011 - 0.9664458653i\)
\(L(\frac12)\) \(\approx\) \(0.1232281011 - 0.9664458653i\)
\(L(1)\) \(\approx\) \(0.9321953586 - 0.3793717900i\)
\(L(1)\) \(\approx\) \(0.9321953586 - 0.3793717900i\)

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

Imaginary part of the first few zeros on the critical line

−18.65311058816577303870946333696, −17.95560708184952464111495927482, −17.217802449567285713649360930702, −16.948647638311471364052663466421, −15.95389591340815641639318278887, −15.5860946580157779152589748312, −14.64005191737883814552461580726, −14.136130899329667033266105274906, −13.43396320412620246269477406811, −12.68920348409245237935086105655, −11.8246025817198604982215235338, −11.08450047143178849418487311884, −10.42849154517598364791185961322, −9.70668320590486399069879908665, −9.359945981939186684211257708, −8.49108411707454178139139465196, −7.81015499322568421181616811390, −6.56542030500806139122680124861, −6.007518300865776691693924328491, −5.33422937779380761157107418683, −4.63364916699245694377339664406, −3.85369929667617433584881924396, −3.00901214052047250076575230153, −2.161056810169186710546846379573, −1.24758816755468176385885257785, 0.25408298814772151498000353461, 1.532644730960614451504255990744, 1.936985582609901617533551166849, 2.885537938780570033862822415446, 3.58700215084853657775037766411, 5.01629867016488676711539545231, 5.48838664406386486742528876755, 6.199106750816951834151545592326, 7.03962300736602861807995616227, 7.39073464576051615037945647522, 8.43585821373477218900205865257, 9.11142197378548367452810207663, 9.80307797722375454943783234189, 10.679159104189461871485624456292, 11.285288933602426326293682588082, 12.26052680160611786896933698184, 12.54302481137397691154468761039, 13.69280169502218294340292702714, 13.82914409146205937639429925765, 14.43361772437196672314356445789, 15.44069621164274615940101083160, 16.35502248274373677317032796233, 16.88569713375109174879387485057, 17.729680155930915226351758565705, 18.229506826765523511950427878063

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