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Properties

Label 1-4004-4004.1039-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.520 - 0.853i$
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  + (0.669 − 0.743i)3-s + (0.913 − 0.406i)5-s + (−0.104 − 0.994i)9-s + (0.309 − 0.951i)15-s + (0.104 − 0.994i)17-s + (0.978 + 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.913 − 0.406i)31-s + (−0.669 − 0.743i)37-s + (0.309 + 0.951i)41-s − 43-s + (−0.5 − 0.866i)45-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (0.913 − 0.406i)5-s + (−0.104 − 0.994i)9-s + (0.309 − 0.951i)15-s + (0.104 − 0.994i)17-s + (0.978 + 0.207i)19-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.913 − 0.406i)31-s + (−0.669 − 0.743i)37-s + (0.309 + 0.951i)41-s − 43-s + (−0.5 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.335841052 - 2.378474071i\)
\(L(\frac12)\) \(\approx\) \(1.335841052 - 2.378474071i\)
\(L(1)\) \(\approx\) \(1.389386941 - 0.7513937231i\)
\(L(1)\) \(\approx\) \(1.389386941 - 0.7513937231i\)

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

−18.811776596059733977616631716959, −17.945453857797536658114265611182, −17.37251426675949883253865414841, −16.61492848302539527277531280758, −15.98095406744807360548850747961, −15.11575675721510914364431634040, −14.72749369948693844071670850745, −13.92302467758279489084606176660, −13.491353594216701161763997093664, −12.75727481091883849778555084380, −11.73791049222586238065795228375, −10.873126218159167347824845735, −10.31922209194100070633512956061, −9.80336504807575197352642144438, −8.93726474449652125742064258352, −8.60466092990497669073582920669, −7.40865389206175192811226180876, −6.97051534844279226055114400803, −5.72233165166507523956887659951, −5.3852015188676152915225099747, −4.45844226541932826001970233184, −3.39538101360776848979286158924, −3.10004006433652673634996133093, −2.00205193940519344303174656433, −1.401141605105048023922112408433, 0.66770083087550503438582009667, 1.40371893414752088544014164237, 2.34163468294193862421089210526, 2.83339867313379547162897207802, 3.82691083003133403604560503102, 4.83336068529208246118893201355, 5.6054580272618631132219567348, 6.30776645556735104670212103505, 7.10646076973597136909155214265, 7.71103113155518197615099449446, 8.59390600857570704376947408625, 9.20685824671137645329594270615, 9.709495162420122113995133518079, 10.53810489807981505773481037799, 11.600619481615487002865954601610, 12.20109271963525132269426875615, 12.9108009210634936816627962328, 13.5623570681290179041018906790, 13.98493431592200757835717655509, 14.65925733390090765103943438907, 15.42400720314441067634897872638, 16.39355257410938369365009321269, 16.863708049628298675357784900803, 17.91394871393052361358973915838, 18.11274015771581138488234308665

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