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Properties

Label 1-4004-4004.1059-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.913 + 0.407i$
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  + (0.809 − 0.587i)3-s + (−0.743 − 0.669i)5-s + (0.309 − 0.951i)9-s + (−0.994 − 0.104i)15-s + (−0.669 + 0.743i)17-s + (−0.587 − 0.809i)19-s + (−0.5 + 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.951i)27-s + (−0.104 + 0.994i)29-s + (−0.743 + 0.669i)31-s + (0.406 + 0.913i)37-s + (0.406 − 0.913i)41-s + (−0.5 + 0.866i)43-s + (−0.866 + 0.5i)45-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.743 − 0.669i)5-s + (0.309 − 0.951i)9-s + (−0.994 − 0.104i)15-s + (−0.669 + 0.743i)17-s + (−0.587 − 0.809i)19-s + (−0.5 + 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.951i)27-s + (−0.104 + 0.994i)29-s + (−0.743 + 0.669i)31-s + (0.406 + 0.913i)37-s + (0.406 − 0.913i)41-s + (−0.5 + 0.866i)43-s + (−0.866 + 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.913 + 0.407i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.913 + 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324900198 + 0.2823000469i\)
\(L(\frac12)\) \(\approx\) \(1.324900198 + 0.2823000469i\)
\(L(1)\) \(\approx\) \(1.063908927 - 0.2220730496i\)
\(L(1)\) \(\approx\) \(1.063908927 - 0.2220730496i\)

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

−18.54444971497748356265059711698, −17.98997574418781146638734594031, −16.82619092905009447936695814331, −16.270425180485751237427321139043, −15.61437314477159205670473393414, −15.01954937461352392862558152480, −14.45394223325994879253304306220, −13.89443452730580090820133495473, −13.03447132717868337480754174235, −12.28929418305014739714749238407, −11.318424101899436213207546532301, −10.91240416124236148801235969366, −10.063188033290366891665215696403, −9.51119486403450527782941056711, −8.57720382272126440043262656361, −8.0435155747876453734814267180, −7.3879741924198555124235859173, −6.60420601266656392395458313832, −5.71275023278961170426514141549, −4.603538209084324247383442849970, −4.07714662680414806745581958398, −3.46208655402723339068124180115, −2.485243703109793683581126888081, −2.06425810901058114219546110963, −0.37851040225963133087784584149, 0.93174858301568220142388100051, 1.72302194481182557727698660999, 2.57239551387010571687521424800, 3.563686746334025961528359250897, 4.056783284858496923778779044, 4.9397979358384143385228741425, 5.86960693207147516258519500712, 6.87411972776919369787348137974, 7.31352733714741765618506616391, 8.20735527510659374599830242791, 8.708959886226140344636923397664, 9.18589889675568776644610529819, 10.17403647764547906633716946192, 11.08932246115855539763485539255, 11.79311782911824884545719618546, 12.50880393630346317970656862597, 13.07791583553932815362190558978, 13.564870235085006306605982436033, 14.53036864181367501875052661462, 15.13607132577140152700291470663, 15.65174185844335282428008697128, 16.42967796245717654243130319537, 17.25979225844081926422368471411, 17.902151688879982791267675736471, 18.62369240273938680548444854981

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