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Properties

Label 1-4008-4008.1043-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.899 - 0.436i$
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  + (−0.862 − 0.505i)5-s + (−0.954 + 0.298i)7-s + (0.644 − 0.764i)11-s + (−0.387 − 0.922i)13-s + (−0.982 + 0.188i)17-s + (0.776 − 0.629i)19-s + (0.584 − 0.811i)23-s + (0.489 + 0.872i)25-s + (−0.584 − 0.811i)29-s + (0.843 + 0.537i)31-s + (0.974 + 0.225i)35-s + (0.614 + 0.788i)37-s + (0.421 + 0.906i)41-s + (0.521 − 0.853i)43-s + (0.672 + 0.739i)47-s + ⋯
L(s)  = 1  + (−0.862 − 0.505i)5-s + (−0.954 + 0.298i)7-s + (0.644 − 0.764i)11-s + (−0.387 − 0.922i)13-s + (−0.982 + 0.188i)17-s + (0.776 − 0.629i)19-s + (0.584 − 0.811i)23-s + (0.489 + 0.872i)25-s + (−0.584 − 0.811i)29-s + (0.843 + 0.537i)31-s + (0.974 + 0.225i)35-s + (0.614 + 0.788i)37-s + (0.421 + 0.906i)41-s + (0.521 − 0.853i)43-s + (0.672 + 0.739i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286398882 - 0.2953243906i\)
\(L(\frac12)\) \(\approx\) \(1.286398882 - 0.2953243906i\)
\(L(1)\) \(\approx\) \(0.8098351368 - 0.1421098956i\)
\(L(1)\) \(\approx\) \(0.8098351368 - 0.1421098956i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.862 - 0.505i)T \)
7 \( 1 + (-0.954 + 0.298i)T \)
11 \( 1 + (0.644 - 0.764i)T \)
13 \( 1 + (-0.387 - 0.922i)T \)
17 \( 1 + (-0.982 + 0.188i)T \)
19 \( 1 + (0.776 - 0.629i)T \)
23 \( 1 + (0.584 - 0.811i)T \)
29 \( 1 + (-0.584 - 0.811i)T \)
31 \( 1 + (0.843 + 0.537i)T \)
37 \( 1 + (0.614 + 0.788i)T \)
41 \( 1 + (0.421 + 0.906i)T \)
43 \( 1 + (0.521 - 0.853i)T \)
47 \( 1 + (0.672 + 0.739i)T \)
53 \( 1 + (-0.351 - 0.936i)T \)
59 \( 1 + (-0.982 - 0.188i)T \)
61 \( 1 + (0.914 + 0.404i)T \)
67 \( 1 + (-0.862 + 0.505i)T \)
71 \( 1 + (-0.280 + 0.959i)T \)
73 \( 1 + (0.999 + 0.0378i)T \)
79 \( 1 + (-0.993 + 0.113i)T \)
83 \( 1 + (-0.700 - 0.713i)T \)
89 \( 1 + (-0.997 + 0.0756i)T \)
97 \( 1 + (-0.843 + 0.537i)T \)
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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

−18.45318814144997728763344655707, −17.720513683670511581475747044003, −16.839658290653504201336675942392, −16.37409431924262211887283368251, −15.51988360283719135787575203560, −15.15675073093877429395687681334, −14.18180603680959940128372469680, −13.77012837926512430399287254569, −12.66657589079045480461595255895, −12.26876041320804291188984933549, −11.417624304097761046975544682, −10.95343242964161233716195961831, −9.92386381519855522437794587156, −9.40771761091957336021091994257, −8.77893771585197001256494724801, −7.47406512799809818200827633915, −7.275282183391596282531574102589, −6.576228970707639090864270094843, −5.77645899356574923923779510856, −4.53644551753315595435956634658, −4.105373292188886185044197130215, −3.32755219610221180055281868981, −2.5404267262978125503214539041, −1.53204272866359714734748077936, −0.40932109251123821535904357714, 0.501638211347998247823344342665, 1.05390437331148484316715213393, 2.60047250652514998139149750885, 3.07254631641204505132131620841, 3.96263954698350567721155491756, 4.660399906893540801593426729815, 5.53122624942384205508612512648, 6.331913236722248079458058614343, 6.99289147472519015564434318681, 7.84892465679651960825233141539, 8.59761791998915277242948708242, 9.11837079641148777049745398442, 9.86091687738355974619989045018, 10.77975685074252990943856240866, 11.49904597293857830736605459101, 12.029849273675647824653942288021, 12.97857073208837258978852851068, 13.13846769203490416769339706539, 14.18841591001509086556282471935, 15.08103588930325694513427121287, 15.64423668561438767692766294226, 16.07001288219929827235366659052, 16.91441681305379140616726902771, 17.39362222054819625550466951150, 18.4135343561809010256705756548

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