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Properties

Label 1-4008-4008.2411-r1-0-0
Degree $1$
Conductor $4008$
Sign $-0.418 + 0.908i$
Analytic cond. $430.719$
Root an. cond. $430.719$
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  + (−0.974 − 0.225i)5-s + (0.0944 − 0.995i)7-s + (−0.997 − 0.0756i)11-s + (−0.169 − 0.985i)13-s + (0.862 − 0.505i)17-s + (0.822 + 0.569i)19-s + (0.881 + 0.472i)23-s + (0.898 + 0.438i)25-s + (−0.881 + 0.472i)29-s + (0.0189 − 0.999i)31-s + (−0.316 + 0.948i)35-s + (0.280 + 0.959i)37-s + (−0.999 − 0.0378i)41-s + (−0.614 − 0.788i)43-s + (0.132 − 0.991i)47-s + ⋯
L(s)  = 1  + (−0.974 − 0.225i)5-s + (0.0944 − 0.995i)7-s + (−0.997 − 0.0756i)11-s + (−0.169 − 0.985i)13-s + (0.862 − 0.505i)17-s + (0.822 + 0.569i)19-s + (0.881 + 0.472i)23-s + (0.898 + 0.438i)25-s + (−0.881 + 0.472i)29-s + (0.0189 − 0.999i)31-s + (−0.316 + 0.948i)35-s + (0.280 + 0.959i)37-s + (−0.999 − 0.0378i)41-s + (−0.614 − 0.788i)43-s + (0.132 − 0.991i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (2411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ -0.418 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1172396338 - 0.1831003939i\)
\(L(\frac12)\) \(\approx\) \(-0.1172396338 - 0.1831003939i\)
\(L(1)\) \(\approx\) \(0.7555692378 - 0.2296108471i\)
\(L(1)\) \(\approx\) \(0.7555692378 - 0.2296108471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.974 - 0.225i)T \)
7 \( 1 + (0.0944 - 0.995i)T \)
11 \( 1 + (-0.997 - 0.0756i)T \)
13 \( 1 + (-0.169 - 0.985i)T \)
17 \( 1 + (0.862 - 0.505i)T \)
19 \( 1 + (0.822 + 0.569i)T \)
23 \( 1 + (0.881 + 0.472i)T \)
29 \( 1 + (-0.881 + 0.472i)T \)
31 \( 1 + (0.0189 - 0.999i)T \)
37 \( 1 + (0.280 + 0.959i)T \)
41 \( 1 + (-0.999 - 0.0378i)T \)
43 \( 1 + (-0.614 - 0.788i)T \)
47 \( 1 + (0.132 - 0.991i)T \)
53 \( 1 + (-0.929 + 0.369i)T \)
59 \( 1 + (0.862 + 0.505i)T \)
61 \( 1 + (0.752 - 0.658i)T \)
67 \( 1 + (-0.974 + 0.225i)T \)
71 \( 1 + (0.700 + 0.713i)T \)
73 \( 1 + (-0.206 - 0.978i)T \)
79 \( 1 + (-0.584 + 0.811i)T \)
83 \( 1 + (-0.942 - 0.334i)T \)
89 \( 1 + (0.914 + 0.404i)T \)
97 \( 1 + (-0.0189 - 0.999i)T \)
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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

−18.8791943799522544249145383393, −18.27636679849337053367499308595, −17.51567671411557934428909812435, −16.48780450009840212381189740053, −16.04227916370625991221519164282, −15.39858535643302608022783530360, −14.738876820084089273921953734768, −14.25361906962975150442584230597, −13.091629137608122767052936144024, −12.60524861935105512088842525702, −11.80202142889242335684647957906, −11.37107667074060021979366630027, −10.62166721493669463892384839255, −9.72252769597340689081528884193, −8.98401294033531528461168231637, −8.298839290557516689949925582, −7.617480913766071635837861612053, −6.97813119690139404936251757936, −6.09618307138361413435364594214, −5.15524278105474543272116107473, −4.72668185733533555838535375690, −3.619839572020764638442583559520, −2.96818321055773552428868061257, −2.21599948678631468551325635478, −1.14849055129314572448915250585, 0.047203795766260287169909063249, 0.68783425265121754688549072423, 1.53344952139368281710958972988, 3.00231652716213228068291503102, 3.34608975038658522526272693444, 4.205629579085541719070284825256, 5.22708197845129585098992609885, 5.431370850682246268969057204107, 6.88007206554915547565774708199, 7.50224125488806846960829570899, 7.87570881143778262280651896647, 8.57122809082406112816067280499, 9.755958580848773997650215557732, 10.19415823117715957381774317116, 11.019720964271602629131493673290, 11.5904010612303435522612537669, 12.36808057903909384069855785716, 13.1511832311404876443675946446, 13.56060157984099395439399266902, 14.57134960359553374628363226334, 15.20177166783484623262280910578, 15.765790318812292839011096220949, 16.64469897431180049232084883869, 16.88861046989636516529941099951, 17.91684292820291285441398173707

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