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Properties

Label 1-4008-4008.773-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: $\chi_{4008} (773, \cdot )$
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

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

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