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Properties

Label 1-4004-4004.1315-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.204 + 0.978i$
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  + (−0.978 − 0.207i)3-s + (0.587 − 0.809i)5-s + (0.913 + 0.406i)9-s + (−0.743 + 0.669i)15-s + (0.104 − 0.994i)17-s + (0.743 + 0.669i)19-s + (−0.5 + 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.669 − 0.743i)29-s + (0.587 + 0.809i)31-s + (−0.743 + 0.669i)37-s + (−0.207 + 0.978i)41-s + (0.5 + 0.866i)43-s + (0.866 − 0.5i)45-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)3-s + (0.587 − 0.809i)5-s + (0.913 + 0.406i)9-s + (−0.743 + 0.669i)15-s + (0.104 − 0.994i)17-s + (0.743 + 0.669i)19-s + (−0.5 + 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (−0.669 − 0.743i)29-s + (0.587 + 0.809i)31-s + (−0.743 + 0.669i)37-s + (−0.207 + 0.978i)41-s + (0.5 + 0.866i)43-s + (0.866 − 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3170772077 + 0.3900383971i\)
\(L(\frac12)\) \(\approx\) \(0.3170772077 + 0.3900383971i\)
\(L(1)\) \(\approx\) \(0.7630247198 - 0.08339792274i\)
\(L(1)\) \(\approx\) \(0.7630247198 - 0.08339792274i\)

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

−18.123145842804500601977782714207, −17.63689115242131364703222399451, −17.04316313800752432920778561365, −16.349717380089565279118862891458, −15.569212546383732315323818009306, −14.98028034819528305796027094229, −14.23955159944318570386771359374, −13.48279260159832155089441586489, −12.7411350927852386817150018137, −12.06245985502640225505004482653, −11.26813658673506845396681007768, −10.69047543889293717647233604235, −10.17741854122364726253154032729, −9.48690167891238380426001746963, −8.65633538527190343021682448771, −7.52140658769054643202397783612, −6.9505407931308819749486422706, −6.20659367798867423827719569157, −5.678115409690325712667084646828, −4.93285880415902003627879715337, −3.99175569968573563074569841068, −3.27781153737497575636654325569, −2.200289499270774719479440401223, −1.4553297549567882260689704162, −0.16659917534429346773797980448, 1.15686637879617372453796841978, 1.55251173069783277500309298222, 2.70480826900530841534937572468, 3.78043353941674188961194558365, 4.815326236382001778668800261594, 5.12909110042964316181708083010, 5.9912797993642405734753056380, 6.484448309618987246872624518560, 7.56329188095030269817557745725, 8.014002585265159804678170838185, 9.18791464250729862613654246215, 9.73227583696649644405074239754, 10.283734357268618999298827002505, 11.29967151282183355273810934767, 11.881120103986607978137668588390, 12.35778552984179750709555850872, 13.275486854438194529744575509926, 13.63744589651470944470316845069, 14.462241357222095279820249396562, 15.613860174404369549170139747003, 16.12044139024400611835045328597, 16.6154475702426184862445791366, 17.4256979946714336178046003574, 17.81154423115833597037048740545, 18.49885638019689359895964864626

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