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Properties

Label 1-4008-4008.1259-r1-0-0
Degree $1$
Conductor $4008$
Sign $-0.307 + 0.951i$
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  + (−0.726 + 0.686i)5-s + (−0.206 + 0.978i)7-s + (0.700 + 0.713i)11-s + (−0.843 − 0.537i)13-s + (−0.752 − 0.658i)17-s + (0.997 + 0.0756i)19-s + (−0.898 − 0.438i)23-s + (0.0567 − 0.998i)25-s + (0.898 − 0.438i)29-s + (−0.553 − 0.832i)31-s + (−0.521 − 0.853i)35-s + (−0.584 − 0.811i)37-s + (−0.387 + 0.922i)41-s + (0.993 + 0.113i)43-s + (0.822 + 0.569i)47-s + ⋯
L(s)  = 1  + (−0.726 + 0.686i)5-s + (−0.206 + 0.978i)7-s + (0.700 + 0.713i)11-s + (−0.843 − 0.537i)13-s + (−0.752 − 0.658i)17-s + (0.997 + 0.0756i)19-s + (−0.898 − 0.438i)23-s + (0.0567 − 0.998i)25-s + (0.898 − 0.438i)29-s + (−0.553 − 0.832i)31-s + (−0.521 − 0.853i)35-s + (−0.584 − 0.811i)37-s + (−0.387 + 0.922i)41-s + (0.993 + 0.113i)43-s + (0.822 + 0.569i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.307 + 0.951i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ -0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6712739481 + 0.9219706443i\)
\(L(\frac12)\) \(\approx\) \(0.6712739481 + 0.9219706443i\)
\(L(1)\) \(\approx\) \(0.8219717108 + 0.1952480204i\)
\(L(1)\) \(\approx\) \(0.8219717108 + 0.1952480204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.726 + 0.686i)T \)
7 \( 1 + (-0.206 + 0.978i)T \)
11 \( 1 + (0.700 + 0.713i)T \)
13 \( 1 + (-0.843 - 0.537i)T \)
17 \( 1 + (-0.752 - 0.658i)T \)
19 \( 1 + (0.997 + 0.0756i)T \)
23 \( 1 + (-0.898 - 0.438i)T \)
29 \( 1 + (0.898 - 0.438i)T \)
31 \( 1 + (-0.553 - 0.832i)T \)
37 \( 1 + (-0.584 - 0.811i)T \)
41 \( 1 + (-0.387 + 0.922i)T \)
43 \( 1 + (0.993 + 0.113i)T \)
47 \( 1 + (0.822 + 0.569i)T \)
53 \( 1 + (-0.672 + 0.739i)T \)
59 \( 1 + (-0.752 + 0.658i)T \)
61 \( 1 + (-0.954 - 0.298i)T \)
67 \( 1 + (-0.726 - 0.686i)T \)
71 \( 1 + (0.881 + 0.472i)T \)
73 \( 1 + (0.169 - 0.985i)T \)
79 \( 1 + (0.489 + 0.872i)T \)
83 \( 1 + (0.421 - 0.906i)T \)
89 \( 1 + (0.942 - 0.334i)T \)
97 \( 1 + (0.553 - 0.832i)T \)
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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

−17.96539323310262831966246504253, −17.24179767987035209105791198359, −16.79169763947055024018698496502, −16.04961032575384887417959265911, −15.647900963865526958005943225766, −14.598911892612927979767295665539, −13.9009483865376543231813789723, −13.51063684178209899713993668477, −12.38594848438284121852092909736, −12.08707054209020621490165948225, −11.24185880348945579309822517625, −10.597725485656498915216782648529, −9.73066711929534866347470237012, −9.01146962663451471379582923929, −8.39763731027915829789911854699, −7.53299898661688767884762879647, −6.995674838876960805765027158076, −6.21063034681810611782435981260, −5.16091258405740741167346804008, −4.5082230431447963418379386434, −3.75937880504660679005301297475, −3.27071361933786948189101720962, −1.89091597311734471761002783352, −1.09235465047514810825539983077, −0.27794660344887046465222172665, 0.59603313347541308793436569622, 1.99692086418602072217446106133, 2.6260728492653337310734887271, 3.318190234421237674361451588268, 4.31936893746431076430358716057, 4.8934908269476317922593351018, 5.97674899351369425649737729572, 6.50747837901793654447025969820, 7.53064129820287380767096011618, 7.73859561319681743030813396825, 8.940736777468060014645016775906, 9.44329090007350863970111963804, 10.19825468444133149711568956646, 11.01218938076010853860603206827, 11.95493396483045260794081092900, 12.03574412429916610968670571848, 12.84701393295289509048027784657, 14.01355142820373653264278091517, 14.41645111332141217598319157356, 15.353785032383593223033501433252, 15.52974355732968528707818155673, 16.31787443356642199489920222586, 17.27811216997951524485888393118, 18.04360443273285288325226586782, 18.369718885043692040316338665491

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