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Properties

Label 1-4008-4008.245-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.448 + 0.893i$
Analytic cond. $18.6130$
Root an. cond. $18.6130$
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  + (−0.898 + 0.438i)5-s + (−0.982 + 0.188i)7-s + (0.988 − 0.150i)11-s + (−0.942 − 0.334i)13-s + (0.489 + 0.872i)17-s + (−0.351 + 0.936i)19-s + (0.553 − 0.832i)23-s + (0.614 − 0.788i)25-s + (0.553 + 0.832i)29-s + (−0.999 + 0.0378i)31-s + (0.800 − 0.599i)35-s + (−0.843 − 0.537i)37-s + (0.997 − 0.0756i)41-s + (−0.243 − 0.969i)43-s + (0.965 − 0.261i)47-s + ⋯
L(s)  = 1  + (−0.898 + 0.438i)5-s + (−0.982 + 0.188i)7-s + (0.988 − 0.150i)11-s + (−0.942 − 0.334i)13-s + (0.489 + 0.872i)17-s + (−0.351 + 0.936i)19-s + (0.553 − 0.832i)23-s + (0.614 − 0.788i)25-s + (0.553 + 0.832i)29-s + (−0.999 + 0.0378i)31-s + (0.800 − 0.599i)35-s + (−0.843 − 0.537i)37-s + (0.997 − 0.0756i)41-s + (−0.243 − 0.969i)43-s + (0.965 − 0.261i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8193457617 + 0.5057821562i\)
\(L(\frac12)\) \(\approx\) \(0.8193457617 + 0.5057821562i\)
\(L(1)\) \(\approx\) \(0.7912267677 + 0.1125033496i\)
\(L(1)\) \(\approx\) \(0.7912267677 + 0.1125033496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.898 + 0.438i)T \)
7 \( 1 + (-0.982 + 0.188i)T \)
11 \( 1 + (0.988 - 0.150i)T \)
13 \( 1 + (-0.942 - 0.334i)T \)
17 \( 1 + (0.489 + 0.872i)T \)
19 \( 1 + (-0.351 + 0.936i)T \)
23 \( 1 + (0.553 - 0.832i)T \)
29 \( 1 + (0.553 + 0.832i)T \)
31 \( 1 + (-0.999 + 0.0378i)T \)
37 \( 1 + (-0.843 - 0.537i)T \)
41 \( 1 + (0.997 - 0.0756i)T \)
43 \( 1 + (-0.243 - 0.969i)T \)
47 \( 1 + (0.965 - 0.261i)T \)
53 \( 1 + (-0.726 - 0.686i)T \)
59 \( 1 + (-0.489 + 0.872i)T \)
61 \( 1 + (-0.132 - 0.991i)T \)
67 \( 1 + (0.898 + 0.438i)T \)
71 \( 1 + (-0.0189 - 0.999i)T \)
73 \( 1 + (0.914 + 0.404i)T \)
79 \( 1 + (0.316 - 0.948i)T \)
83 \( 1 + (-0.776 + 0.629i)T \)
89 \( 1 + (-0.672 + 0.739i)T \)
97 \( 1 + (-0.999 - 0.0378i)T \)
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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

−18.59808482776612848365156772834, −17.35893388488847168080484034174, −17.04861734448106353140080570463, −16.30366754312133428855292646402, −15.68164974595226650624582823614, −15.10708087884679357090256191613, −14.25658166023561863479053504056, −13.59302594360245369795642930099, −12.67283479933046148349208587827, −12.28803207749488386298286361890, −11.52903604649181574706673140915, −10.98271372074729634857782685217, −9.75611179184252435674080648502, −9.42141117257625156133202534860, −8.78152384063625097831786684507, −7.71094878708246954998281471860, −7.13759585756809835999874138762, −6.6247911968318728129267625164, −5.57072380003730502461668026824, −4.69186174649530236406721069151, −4.128814963858478513600219359425, −3.280774620448313606524795779642, −2.61474776411065595709018644491, −1.332592957687618172111409611973, −0.42840821850935953974172474116, 0.68888302383454180964044105040, 1.92395895384009690739699816311, 2.894013422649839890460098117975, 3.62009748260519368859872410924, 4.051775411873228950918577142417, 5.165367545255100843856343865402, 6.02690634531329641764877746676, 6.757280196667925181459939385095, 7.26194930814072512615359366685, 8.168151533781991656729934875278, 8.84659557976242947980467624124, 9.59376279613486162407265436074, 10.488505867373352876650395915397, 10.828821517959495735330538248570, 12.00243698769364861343699766384, 12.41155164986904379607258746287, 12.77638139762087528358697243185, 14.12553686174924670787682226247, 14.52306817221041022240160067086, 15.16568550192883623184364958320, 15.8629016082137308618424017013, 16.70377353434687052755487178932, 16.93629078718123748569076097206, 18.02435702468414937254824415237, 18.87300646897519048651358461723

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