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Properties

Label 1-4004-4004.135-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.226 - 0.973i$
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  + (0.104 + 0.994i)3-s + (−0.743 − 0.669i)5-s + (−0.978 + 0.207i)9-s + (0.587 − 0.809i)15-s + (0.978 + 0.207i)17-s + (−0.406 + 0.913i)19-s + (−0.5 − 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (−0.743 + 0.669i)31-s + (−0.994 − 0.104i)37-s + (0.587 + 0.809i)41-s + 43-s + (0.866 + 0.5i)45-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)3-s + (−0.743 − 0.669i)5-s + (−0.978 + 0.207i)9-s + (0.587 − 0.809i)15-s + (0.978 + 0.207i)17-s + (−0.406 + 0.913i)19-s + (−0.5 − 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (−0.743 + 0.669i)31-s + (−0.994 − 0.104i)37-s + (0.587 + 0.809i)41-s + 43-s + (0.866 + 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.226 - 0.973i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.226 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4603975086 - 0.3654693557i\)
\(L(\frac12)\) \(\approx\) \(0.4603975086 - 0.3654693557i\)
\(L(1)\) \(\approx\) \(0.7930576551 + 0.1400805034i\)
\(L(1)\) \(\approx\) \(0.7930576551 + 0.1400805034i\)

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

−18.79720507933239577246676278285, −17.94332290370425127265203497433, −17.54718044056717815569820500878, −16.58916292150972999736755772644, −15.922381417919714528152521158158, −15.010030837727509705915891190, −14.60110186686960336549063181982, −13.80689412257636707699979046214, −13.21332184679859400296073939497, −12.29772134432314588753268672583, −11.93257714060076771118818826955, −11.09449055410537569061389195663, −10.626958023240599036179941339472, −9.47955849002633475872437429754, −8.80544206789161631322764558299, −7.930230428607483235294996872114, −7.34405597498797589928772810470, −7.00038348714167137109325342173, −5.95609975952835641090874225470, −5.4265379640692905177307921676, −4.18640251130257346629354538583, −3.45076144438663518800445176537, −2.727043321225113394590997718137, −1.93567839204057507564775183573, −0.90061405674803342494251265051, 0.19431890171809366570284874366, 1.44650053630919853325588843431, 2.516955217305319398176462342354, 3.621773533914848711644696760176, 3.89476295923366663281768685962, 4.75200347909088388107135881412, 5.4771327123836584204963166819, 6.10525932093236523025187399934, 7.33980467404775787651989842919, 8.04302296412211311984510361274, 8.59789885900184609248117147189, 9.296999065655556657124914190848, 10.060195706040835141776971378052, 10.690461864638800840116823390836, 11.40104187401932405933386597670, 12.25121494965601356465439479889, 12.58947631970182024629453577981, 13.7006582277315266609737059385, 14.48451124305871877504691111633, 14.92485049374651128431465642480, 15.743614302668733508538394277943, 16.29829303109253118649343433203, 16.77298357209649273328113965053, 17.36642590268994236973810458374, 18.483098443587809130629199620568

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