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Properties

Label 1-4008-4008.1037-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.201 - 0.979i$
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  + (−0.206 − 0.978i)5-s + (−0.843 − 0.537i)7-s + (0.898 + 0.438i)11-s + (−0.521 + 0.853i)13-s + (−0.999 + 0.0378i)17-s + (0.881 + 0.472i)19-s + (−0.982 + 0.188i)23-s + (−0.914 + 0.404i)25-s + (−0.982 − 0.188i)29-s + (−0.993 − 0.113i)31-s + (−0.351 + 0.936i)35-s + (0.132 + 0.991i)37-s + (0.974 + 0.225i)41-s + (0.672 − 0.739i)43-s + (0.700 + 0.713i)47-s + ⋯
L(s)  = 1  + (−0.206 − 0.978i)5-s + (−0.843 − 0.537i)7-s + (0.898 + 0.438i)11-s + (−0.521 + 0.853i)13-s + (−0.999 + 0.0378i)17-s + (0.881 + 0.472i)19-s + (−0.982 + 0.188i)23-s + (−0.914 + 0.404i)25-s + (−0.982 − 0.188i)29-s + (−0.993 − 0.113i)31-s + (−0.351 + 0.936i)35-s + (0.132 + 0.991i)37-s + (0.974 + 0.225i)41-s + (0.672 − 0.739i)43-s + (0.700 + 0.713i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8147406716 - 0.6639483519i\)
\(L(\frac12)\) \(\approx\) \(0.8147406716 - 0.6639483519i\)
\(L(1)\) \(\approx\) \(0.8533551450 - 0.1663229962i\)
\(L(1)\) \(\approx\) \(0.8533551450 - 0.1663229962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.206 - 0.978i)T \)
7 \( 1 + (-0.843 - 0.537i)T \)
11 \( 1 + (0.898 + 0.438i)T \)
13 \( 1 + (-0.521 + 0.853i)T \)
17 \( 1 + (-0.999 + 0.0378i)T \)
19 \( 1 + (0.881 + 0.472i)T \)
23 \( 1 + (-0.982 + 0.188i)T \)
29 \( 1 + (-0.982 - 0.188i)T \)
31 \( 1 + (-0.993 - 0.113i)T \)
37 \( 1 + (0.132 + 0.991i)T \)
41 \( 1 + (0.974 + 0.225i)T \)
43 \( 1 + (0.672 - 0.739i)T \)
47 \( 1 + (0.700 + 0.713i)T \)
53 \( 1 + (0.644 + 0.764i)T \)
59 \( 1 + (0.999 + 0.0378i)T \)
61 \( 1 + (0.387 - 0.922i)T \)
67 \( 1 + (0.206 - 0.978i)T \)
71 \( 1 + (0.0567 - 0.998i)T \)
73 \( 1 + (0.316 - 0.948i)T \)
79 \( 1 + (-0.822 - 0.569i)T \)
83 \( 1 + (0.455 - 0.890i)T \)
89 \( 1 + (0.800 - 0.599i)T \)
97 \( 1 + (-0.993 + 0.113i)T \)
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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

−18.60187658899539596417893060262, −17.94197233518998142860613537192, −17.5163502348040055010826860047, −16.327599710856915146669619673049, −15.99198574667994000071107211659, −15.16240390454673204488483252384, −14.62673041739363506361572121080, −13.955753084735333818177862220382, −13.108247923191994854666157521390, −12.5005716242527126095051909218, −11.624392595196569041248838377496, −11.16888899108807455695732874227, −10.33761532823108645237483268702, −9.588878191624751449727972406408, −9.056643396830113731286219273994, −8.16246613415679129834757000210, −7.14889492776979913154511466441, −6.89111365064704421213318858770, −5.81670961814443460274133462222, −5.5252755063014219219976355185, −4.02561734925827292786732115956, −3.663403516272460669472858530682, −2.63989576579255324730578918785, −2.2514499739058256130867213071, −0.73531006896709405510414192701, 0.406297305017776567732970452961, 1.50610502815739875298743628638, 2.17953760083920423323213087747, 3.49082448289581198619910302211, 4.10637767080518768400066327728, 4.59329952585147072775785391724, 5.66991027946177834977446244472, 6.33877113176846961225530332249, 7.23039713541550371729611556098, 7.67980772952281873770304364386, 8.80642609047860979281337259929, 9.4405309360357545622097832519, 9.67148482454446681290662314254, 10.78148605830449911019993054546, 11.65033906189043867988369538176, 12.17598443702375671900252030781, 12.798527139777466192656493821869, 13.5737879475289026856141964258, 14.09010067640528934735417212208, 14.974232247077595375424960423000, 15.82118740314134644168089962664, 16.34059968668001321854896252645, 16.9220012310791524159538119358, 17.42684547668080964997736202437, 18.335326080801588301649515086927

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