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Properties

Label 1-4004-4004.1271-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.932 + 0.360i$
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 + (−0.913 + 0.406i)5-s + (−0.809 + 0.587i)9-s + (0.669 + 0.743i)15-s + (−0.913 + 0.406i)17-s + (−0.309 − 0.951i)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 + (0.913 + 0.406i)31-s + (0.978 − 0.207i)37-s + (−0.978 − 0.207i)41-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)45-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.913 + 0.406i)5-s + (−0.809 + 0.587i)9-s + (0.669 + 0.743i)15-s + (−0.913 + 0.406i)17-s + (−0.309 − 0.951i)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 + (0.913 + 0.406i)31-s + (0.978 − 0.207i)37-s + (−0.978 − 0.207i)41-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.932 + 0.360i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.932 + 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02409677851 - 0.1293455568i\)
\(L(\frac12)\) \(\approx\) \(0.02409677851 - 0.1293455568i\)
\(L(1)\) \(\approx\) \(0.6526029836 - 0.1570973726i\)
\(L(1)\) \(\approx\) \(0.6526029836 - 0.1570973726i\)

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

−18.85256952722112229924571575151, −18.19277349375020896046336544210, −17.29270984143929007715276292247, −16.59719072163017409243770190967, −16.25257385757140285012542279483, −15.51572811398378141871588353152, −14.86768073878031783091481257222, −14.4386758780679858538029220540, −13.23752926037514530036214309015, −12.68196502632049847868772092585, −11.70287108471748610212258007564, −11.43041833995060956894729945442, −10.62414146530404294475763108650, −9.914835422246089939653139742769, −9.14454169934751623263603105313, −8.464371596371564999691611085553, −7.925920260918742996861558712464, −6.799975529315913017157253729547, −6.21702219362770579215256732527, −5.07099787770020260266145191723, −4.734962314743897321517882556497, −3.90851865921340590592829749044, −3.317214000867515649690008021288, −2.33202565711692551324082782098, −1.01285574070507316713233738276, 0.04976282087201310910879686053, 1.04321311039661501622661990735, 2.140232578975926683592128057913, 2.79153969323309962265464019052, 3.74087254933341846609616259165, 4.56500205483645388725337189844, 5.41294529275796400687717896950, 6.31483065173770976483646056948, 6.98719960563090983355724872139, 7.377290979242815822631678640541, 8.37752720354916537030465542838, 8.68939285764295702559518935704, 9.88505422686569009963081458619, 10.78943601305861767443779326568, 11.420471048510034921663870269813, 11.73702119568511467200634821387, 12.64662839252074772186533365712, 13.32497783017052193602056629672, 13.75931838619458785359503145508, 14.954557638717476723457487880287, 15.176480150141312688054039553829, 16.08822276697417280197264561460, 16.85666240116119432029349921798, 17.59639988582264876530814670133, 18.02109831053397772714204847329

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