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Properties

Label 1-4004-4004.955-r1-0-0
Degree $1$
Conductor $4004$
Sign $-0.959 + 0.281i$
Analytic cond. $430.289$
Root an. cond. $430.289$
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 + (0.994 − 0.104i)5-s + (−0.104 + 0.994i)9-s + (0.743 + 0.669i)15-s + (−0.809 + 0.587i)17-s + (0.207 + 0.978i)19-s + 23-s + (0.978 − 0.207i)25-s + (−0.809 + 0.587i)27-s + (0.669 − 0.743i)29-s + (0.994 + 0.104i)31-s + (−0.951 + 0.309i)37-s + (−0.207 − 0.978i)41-s + (−0.5 + 0.866i)43-s + i·45-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)3-s + (0.994 − 0.104i)5-s + (−0.104 + 0.994i)9-s + (0.743 + 0.669i)15-s + (−0.809 + 0.587i)17-s + (0.207 + 0.978i)19-s + 23-s + (0.978 − 0.207i)25-s + (−0.809 + 0.587i)27-s + (0.669 − 0.743i)29-s + (0.994 + 0.104i)31-s + (−0.951 + 0.309i)37-s + (−0.207 − 0.978i)41-s + (−0.5 + 0.866i)43-s + i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(430.289\)
Root analytic conductor: \(430.289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (1:\ ),\ -0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3746037283 + 2.611140495i\)
\(L(\frac12)\) \(\approx\) \(0.3746037283 + 2.611140495i\)
\(L(1)\) \(\approx\) \(1.354175228 + 0.6006432900i\)
\(L(1)\) \(\approx\) \(1.354175228 + 0.6006432900i\)

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

−17.96282283567928040126541732560, −17.56646851683993956459249020217, −16.852264945152793513925186259820, −15.83981182939925670887190830053, −15.15163839830096935813565819806, −14.47864368721375195249751279229, −13.70389587674637748201454785341, −13.4047053492882971990233740224, −12.75790511740815681827038958661, −11.89742265524580155363153793487, −11.169731511126703769055105131065, −10.28328301421765026642377889328, −9.54606373003906495683330703287, −8.867191375046933842543146755535, −8.45662405502493607051844785655, −7.26052257005015839409004462354, −6.830402097049255038167179256461, −6.24474144859676984767060018131, −5.1986144206579174678122344281, −4.59198893646203989188136817952, −3.22697051280168853911679301136, −2.812200232287024938191221585688, −1.96989447693891451266386698563, −1.23986597479491073046915286914, −0.30507224479030238337678899124, 1.22722066617298676225339827914, 1.96347233125413131542684936739, 2.792990342348280303100381276338, 3.45621657368731359885298093839, 4.52157790160067277943269257465, 4.939056590993570357625030452118, 5.94867676005363088049362896217, 6.50401904303861107211858966136, 7.59160256988874796942359957965, 8.363400275138632204876882289850, 8.9842299586498959373929427185, 9.57390041363365772402504809047, 10.37613657537760644745131307155, 10.65591514563057873014846416124, 11.716555031476441722074781136628, 12.633291631763117898851214956007, 13.3499857666471339859862249801, 13.916496479617968628295103199018, 14.4381637503840709200147049392, 15.31413082202480851324537688550, 15.70546474315917800517045930590, 16.74376546365266978085060784337, 17.089134457985876369702524212589, 17.85053412629906488664459915641, 18.71927314351052701454655840048

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