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Properties

Label 1-4004-4004.1087-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.567 + 0.823i$
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 + (−0.406 + 0.913i)5-s + (−0.104 + 0.994i)9-s + (0.951 − 0.309i)15-s + (0.104 + 0.994i)17-s + (0.207 + 0.978i)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 + (−0.406 − 0.913i)31-s + (−0.743 − 0.669i)37-s + (0.951 + 0.309i)41-s + 43-s + (−0.866 − 0.5i)45-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)3-s + (−0.406 + 0.913i)5-s + (−0.104 + 0.994i)9-s + (0.951 − 0.309i)15-s + (0.104 + 0.994i)17-s + (0.207 + 0.978i)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 + (−0.406 − 0.913i)31-s + (−0.743 − 0.669i)37-s + (0.951 + 0.309i)41-s + 43-s + (−0.866 − 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.567 + 0.823i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.567 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8905452199 + 0.4676253202i\)
\(L(\frac12)\) \(\approx\) \(0.8905452199 + 0.4676253202i\)
\(L(1)\) \(\approx\) \(0.7848965630 + 0.04185981915i\)
\(L(1)\) \(\approx\) \(0.7848965630 + 0.04185981915i\)

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

−18.09598921131034994283917564834, −17.59880779049835156451934010615, −16.98697591662358807238365019497, −16.22536416306256909204205514163, −15.72614725621807304121025255177, −15.38422657982637620979317797755, −14.27675545914904843696210740702, −13.60513679961132267836037512010, −12.750427679039499198521969412596, −12.01982884174901763614117130560, −11.58791337551412231630738894092, −10.89786995970936646317825574311, −10.02774672522717512593796164395, −9.27730904115073590712634665136, −8.94614550133540884758419438782, −7.88784135228625644403553880187, −7.18588795724052923260103927660, −6.25837799638979066366218820566, −5.39221641789092354542785060085, −4.94737542521711299307641319109, −4.216882011249223739285654565801, −3.51430104747655640782125482542, −2.53989318612396187193630419508, −1.22457455589232200749391610689, −0.45765015871212952521221067849, 0.7788911974312394931543023889, 1.902727549433082892010918639677, 2.48609845586571270603781655883, 3.62369395338698535916701764441, 4.19564039401751922023075791397, 5.40896278490848705877323449841, 5.945650391790897152509176712935, 6.69963016905780196987687873446, 7.25622486756061805636014681178, 8.03425681983583881714239347624, 8.54751893583880213199849352533, 9.884665871675657876426231695840, 10.45361305336377402291767626278, 11.06439014124194145087992055653, 11.68565575597668830560828995444, 12.516643836818246073865354218967, 12.817965809434538580356367738109, 13.97740728361544071828756720829, 14.3835705416259692439821949792, 15.09442106678578983279541522035, 16.08639074503497024584912724365, 16.49057054449076241877837124899, 17.40458634837560710290119982025, 17.96885282181259532160488730655, 18.588637942838792661854896894375

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