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Properties

Label 1-4004-4004.1007-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.981 - 0.191i$
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 + (0.587 − 0.809i)5-s + (−0.104 − 0.994i)9-s + (−0.207 − 0.978i)15-s + (−0.913 + 0.406i)17-s + (0.207 − 0.978i)19-s + (−0.5 − 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.978 − 0.207i)29-s + (0.587 + 0.809i)31-s + (−0.207 − 0.978i)37-s + (−0.743 − 0.669i)41-s + (0.5 − 0.866i)43-s + (−0.866 − 0.5i)45-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (0.587 − 0.809i)5-s + (−0.104 − 0.994i)9-s + (−0.207 − 0.978i)15-s + (−0.913 + 0.406i)17-s + (0.207 − 0.978i)19-s + (−0.5 − 0.866i)23-s + (−0.309 − 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.978 − 0.207i)29-s + (0.587 + 0.809i)31-s + (−0.207 − 0.978i)37-s + (−0.743 − 0.669i)41-s + (0.5 − 0.866i)43-s + (−0.866 − 0.5i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.981 - 0.191i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1826214323 - 1.888428394i\)
\(L(\frac12)\) \(\approx\) \(0.1826214323 - 1.888428394i\)
\(L(1)\) \(\approx\) \(1.105007652 - 0.7306151005i\)
\(L(1)\) \(\approx\) \(1.105007652 - 0.7306151005i\)

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

−18.851283772123409118457682305106, −18.12619513036814159281870107842, −17.48988860179868945699251853400, −16.70302294594350453182160141798, −15.92565697045674664808742274446, −15.36091891580195447184470068181, −14.716393185094941503052602710580, −14.01969887402861950017262363954, −13.62912197654055169617136406078, −12.872947373361155124659550130101, −11.66987838297544714305257557539, −11.194951588261880451184784792521, −10.292405597294205665271320261086, −9.83193838335618546040427164473, −9.33057925443257647804352610874, −8.28526573965078637791093656774, −7.837835528302510150237964289997, −6.79467599502506047285432646562, −6.18914231033230519816653648852, −5.25142421502672457207360413456, −4.55519297242036958031607149186, −3.613452150471762098744096975573, −3.0311026010456908709349769945, −2.25294417464065687365638031652, −1.50056866118258639329931370495, 0.44056861772646192232539213005, 1.33983634056006192871334523877, 2.1891483074641676119269869963, 2.71801142946729656922343785421, 3.84928492335722122742351997853, 4.61632499231359404882249854854, 5.420138956161333159800435157995, 6.43597186406865252378406492894, 6.750323187764915625144913331331, 7.810604469727471473437864496810, 8.570820643255733748630468181947, 8.88377552216858203243413148998, 9.68685003059160640924814753486, 10.47160961732194836340629686397, 11.39614087351711827436409040848, 12.394128653483862319999972172926, 12.582342879116373448068090064644, 13.50171623062247160047540880591, 13.892566061221865718401651018749, 14.53609741142822646246513072511, 15.61782831063537397883098875392, 15.893067896319984210919246709306, 17.11139040189451033421855069908, 17.50519994804767999008551921303, 18.0833317187320985372135461908

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