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Properties

Label 1-4004-4004.1075-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.776 + 0.629i$
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  + (0.104 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.776 + 0.629i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.776 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2401739240 + 0.6779571389i\)
\(L(\frac12)\) \(\approx\) \(0.2401739240 + 0.6779571389i\)
\(L(1)\) \(\approx\) \(0.7585340369 + 0.2502678198i\)
\(L(1)\) \(\approx\) \(0.7585340369 + 0.2502678198i\)

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 \( 1 + (0.669 + 0.743i)T \)
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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

−18.35299226584436431498570097476, −17.62176503881276619430115446405, −17.00619398084996514405534321240, −16.09079848555174949844751299401, −15.49057153074903316595170810036, −14.7761640077015456316518219227, −14.057833059716683835065224221934, −13.450566570460416382454825351595, −12.635632990124332882605836931401, −12.01218986073260811004582904666, −11.58668041320562266649512647867, −10.76149412212189263020611473852, −10.00330490059237564649467955472, −8.87931463917954734995848946598, −8.2740444697053935408817460810, −7.83359176795656722838382910566, −6.81650275666589323428230874319, −6.64028073271269188025247618456, −5.5043404005944762301613948669, −4.739622388785301411290851947434, −3.6227007034957301366404531402, −3.191330590084722237658233596111, −2.13993046447863924855564274480, −1.33657688879891348998998721779, −0.25490794215025044341278846189, 0.862766005078666327460204126820, 2.25475584496860729900970812539, 3.13680891237535184486210839892, 3.75585550924804194932084635992, 4.51026903598047004386012343558, 5.055489624246991189203273594666, 5.92572324415053951087138205264, 6.85666606873882764447630047711, 7.90101713815619129963493426355, 8.14912161849188057807892289197, 9.19651219463034823357198354405, 9.65647119708910957621454376181, 10.47619189184392674424815376794, 11.21222649522393080932596434008, 11.86800472501849079873396184175, 12.24603904512374924968767937054, 13.55020175890477918879346309932, 13.98574090191401992316100458020, 14.83244119523370357527604221079, 15.54333135405111266130620366907, 15.94305467186680398742665077440, 16.45724884422206725839128891075, 17.2712769704493070792097925208, 18.025646073065378393267845737886, 18.86218896146668524767213348670

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