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Properties

Label 1-4008-4008.101-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.997 + 0.0707i$
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  + (−0.988 + 0.150i)5-s + (0.553 + 0.832i)7-s + (−0.455 + 0.890i)11-s + (−0.993 − 0.113i)13-s + (−0.169 − 0.985i)17-s + (0.800 + 0.599i)19-s + (−0.752 − 0.658i)23-s + (0.954 − 0.298i)25-s + (−0.752 + 0.658i)29-s + (0.489 − 0.872i)31-s + (−0.672 − 0.739i)35-s + (−0.982 − 0.188i)37-s + (−0.521 − 0.853i)41-s + (0.822 − 0.569i)43-s + (−0.421 + 0.906i)47-s + ⋯
L(s)  = 1  + (−0.988 + 0.150i)5-s + (0.553 + 0.832i)7-s + (−0.455 + 0.890i)11-s + (−0.993 − 0.113i)13-s + (−0.169 − 0.985i)17-s + (0.800 + 0.599i)19-s + (−0.752 − 0.658i)23-s + (0.954 − 0.298i)25-s + (−0.752 + 0.658i)29-s + (0.489 − 0.872i)31-s + (−0.672 − 0.739i)35-s + (−0.982 − 0.188i)37-s + (−0.521 − 0.853i)41-s + (0.822 − 0.569i)43-s + (−0.421 + 0.906i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.997 + 0.0707i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.997 + 0.0707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056473434 + 0.03741523617i\)
\(L(\frac12)\) \(\approx\) \(1.056473434 + 0.03741523617i\)
\(L(1)\) \(\approx\) \(0.8369010937 + 0.09020913378i\)
\(L(1)\) \(\approx\) \(0.8369010937 + 0.09020913378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.988 + 0.150i)T \)
7 \( 1 + (0.553 + 0.832i)T \)
11 \( 1 + (-0.455 + 0.890i)T \)
13 \( 1 + (-0.993 - 0.113i)T \)
17 \( 1 + (-0.169 - 0.985i)T \)
19 \( 1 + (0.800 + 0.599i)T \)
23 \( 1 + (-0.752 - 0.658i)T \)
29 \( 1 + (-0.752 + 0.658i)T \)
31 \( 1 + (0.489 - 0.872i)T \)
37 \( 1 + (-0.982 - 0.188i)T \)
41 \( 1 + (-0.521 - 0.853i)T \)
43 \( 1 + (0.822 - 0.569i)T \)
47 \( 1 + (-0.421 + 0.906i)T \)
53 \( 1 + (0.700 - 0.713i)T \)
59 \( 1 + (0.169 - 0.985i)T \)
61 \( 1 + (0.843 - 0.537i)T \)
67 \( 1 + (0.988 + 0.150i)T \)
71 \( 1 + (0.862 - 0.505i)T \)
73 \( 1 + (-0.614 + 0.788i)T \)
79 \( 1 + (0.914 - 0.404i)T \)
83 \( 1 + (-0.974 + 0.225i)T \)
89 \( 1 + (0.243 - 0.969i)T \)
97 \( 1 + (0.489 + 0.872i)T \)
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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

−18.51340559003870801876230511500, −17.702053661237019420960061576013, −17.08825138389049897592170121361, −16.43733712913245742337247099603, −15.76793819048455448025556647016, −15.11085721140988175923691906150, −14.43099610657361223027153613637, −13.64791849951743766518043800816, −13.12612109228608196147289155297, −12.095672838629093850134795847212, −11.65256289504549766010012786232, −10.927120081489470023911995117714, −10.33400808280777624220810770853, −9.49303647328089223933552957912, −8.46113794051832413391352750084, −8.03749756425471575247786737362, −7.35750860446103231275332843821, −6.74324595762051057979938633823, −5.57683175065517705211761795842, −4.94758123734228137497314755726, −4.12288712206309319339055469147, −3.554310694921313635373477163731, −2.66530504366306152516004457293, −1.533384087332183170920791760474, −0.62076236880395195037124695952, 0.484648114614932397836009089037, 1.94411831102715764926724947147, 2.474514699344548972061851397970, 3.400288861222659854931726634240, 4.308152655765075952037220214916, 5.04507392137912538968802228911, 5.49958420717799621045742534360, 6.76250833105763460178259474722, 7.396133684128830302071529497041, 7.91839696117325740233149363919, 8.64442547042989356846951049552, 9.54457062150368588732213713937, 10.118999756744495611519240765729, 11.09312336920418551509724237702, 11.6911891604060527459299778120, 12.34594179519962961328170575435, 12.63157417831615470686930710871, 13.93592073265031340453407309075, 14.51876657227519850179901365917, 15.109952006646244253087477093757, 15.770647217804657132716668120290, 16.197969737327209986734929261733, 17.26256986796862043880868567691, 17.861725021076695752938267008576, 18.64272206906366564206294052874

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