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Properties

Label 1-4008-4008.1019-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.996 + 0.0862i$
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  + (−0.421 + 0.906i)5-s + (0.455 + 0.890i)7-s + (−0.929 + 0.369i)11-s + (−0.752 + 0.658i)13-s + (−0.881 + 0.472i)17-s + (−0.993 − 0.113i)19-s + (−0.776 − 0.629i)23-s + (−0.644 − 0.764i)25-s + (0.776 − 0.629i)29-s + (0.0944 + 0.995i)31-s + (−0.999 + 0.0378i)35-s + (0.988 − 0.150i)37-s + (−0.982 + 0.188i)41-s + (0.169 − 0.985i)43-s + (0.614 + 0.788i)47-s + ⋯
L(s)  = 1  + (−0.421 + 0.906i)5-s + (0.455 + 0.890i)7-s + (−0.929 + 0.369i)11-s + (−0.752 + 0.658i)13-s + (−0.881 + 0.472i)17-s + (−0.993 − 0.113i)19-s + (−0.776 − 0.629i)23-s + (−0.644 − 0.764i)25-s + (0.776 − 0.629i)29-s + (0.0944 + 0.995i)31-s + (−0.999 + 0.0378i)35-s + (0.988 − 0.150i)37-s + (−0.982 + 0.188i)41-s + (0.169 − 0.985i)43-s + (0.614 + 0.788i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.996 + 0.0862i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.996 + 0.0862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4170065393 + 0.01802003099i\)
\(L(\frac12)\) \(\approx\) \(0.4170065393 + 0.01802003099i\)
\(L(1)\) \(\approx\) \(0.6776911060 + 0.2602695099i\)
\(L(1)\) \(\approx\) \(0.6776911060 + 0.2602695099i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.421 + 0.906i)T \)
7 \( 1 + (0.455 + 0.890i)T \)
11 \( 1 + (-0.929 + 0.369i)T \)
13 \( 1 + (-0.752 + 0.658i)T \)
17 \( 1 + (-0.881 + 0.472i)T \)
19 \( 1 + (-0.993 - 0.113i)T \)
23 \( 1 + (-0.776 - 0.629i)T \)
29 \( 1 + (0.776 - 0.629i)T \)
31 \( 1 + (0.0944 + 0.995i)T \)
37 \( 1 + (0.988 - 0.150i)T \)
41 \( 1 + (-0.982 + 0.188i)T \)
43 \( 1 + (0.169 - 0.985i)T \)
47 \( 1 + (0.614 + 0.788i)T \)
53 \( 1 + (0.316 - 0.948i)T \)
59 \( 1 + (-0.881 - 0.472i)T \)
61 \( 1 + (-0.898 - 0.438i)T \)
67 \( 1 + (-0.421 - 0.906i)T \)
71 \( 1 + (-0.672 + 0.739i)T \)
73 \( 1 + (-0.862 + 0.505i)T \)
79 \( 1 + (-0.0189 + 0.999i)T \)
83 \( 1 + (0.132 + 0.991i)T \)
89 \( 1 + (-0.489 - 0.872i)T \)
97 \( 1 + (-0.0944 + 0.995i)T \)
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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

−18.106106013321252933992384975957, −17.64210360875275132003953230879, −16.80728236326312139229652117633, −16.44486220033695790772213349305, −15.49845536841293140133628685125, −15.117129052544211848226006443723, −14.11204034709411792634252759704, −13.28728955454593337283158721696, −13.07678833176771361570956489884, −12.07001041248912909618072147652, −11.51016563630464801772496972322, −10.59272721257057640824516245757, −10.19589253205555586029646664359, −9.17174713652738170600705105832, −8.47191662707359221465374712874, −7.719944277827127307202638359077, −7.44162529828319033299663429747, −6.24787138268022889172602804238, −5.444892996123636545516108718195, −4.5739745202820132158178782293, −4.334348014194577991464678050064, −3.20320841149273514968651830435, −2.31995043936633161120723009336, −1.33425350203524442904475025138, −0.39277037890671800599829084349, 0.128580971824125720165555227500, 1.85620490963919527203017874595, 2.363126833185861403426248862255, 2.96043076589750443782008037469, 4.26089052132411989480271571468, 4.59994334182707521684430575613, 5.64996676081205783329872699809, 6.46750086671497908679486740543, 6.98815289426292123164685885753, 7.99928279493949384324981211097, 8.368536826555618704624732070642, 9.292387659005131780575171232265, 10.20115661021908855307587187186, 10.71245148369686380337577249549, 11.430808307110770130254174134565, 12.18398991161242424046014782676, 12.62432803665996934211543245980, 13.675672065843983270566050747042, 14.367175657055849053890689412164, 15.01952755646954104797949513559, 15.45834468133115452903628042491, 16.06444880490223730450648762285, 17.12269861208392454304595054637, 17.769397962708672453004210778501, 18.34797416372347653959477340814

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