Properties

Label 1-4008-4008.1307-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.953 - 0.302i$
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  + (−0.822 + 0.569i)5-s + (0.700 + 0.713i)7-s + (0.316 − 0.948i)11-s + (0.898 + 0.438i)13-s + (0.776 − 0.629i)17-s + (−0.843 + 0.537i)19-s + (0.965 − 0.261i)23-s + (0.351 − 0.936i)25-s + (−0.965 − 0.261i)29-s + (0.455 − 0.890i)31-s + (−0.982 − 0.188i)35-s + (0.726 + 0.686i)37-s + (−0.584 − 0.811i)41-s + (0.752 + 0.658i)43-s + (−0.169 + 0.985i)47-s + ⋯
L(s)  = 1  + (−0.822 + 0.569i)5-s + (0.700 + 0.713i)7-s + (0.316 − 0.948i)11-s + (0.898 + 0.438i)13-s + (0.776 − 0.629i)17-s + (−0.843 + 0.537i)19-s + (0.965 − 0.261i)23-s + (0.351 − 0.936i)25-s + (−0.965 − 0.261i)29-s + (0.455 − 0.890i)31-s + (−0.982 − 0.188i)35-s + (0.726 + 0.686i)37-s + (−0.584 − 0.811i)41-s + (0.752 + 0.658i)43-s + (−0.169 + 0.985i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.953 - 0.302i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.953 - 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.409576124 - 0.3735325939i\)
\(L(\frac12)\) \(\approx\) \(2.409576124 - 0.3735325939i\)
\(L(1)\) \(\approx\) \(1.136879478 + 0.08061801783i\)
\(L(1)\) \(\approx\) \(1.136879478 + 0.08061801783i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.822 + 0.569i)T \)
7 \( 1 + (0.700 + 0.713i)T \)
11 \( 1 + (0.316 - 0.948i)T \)
13 \( 1 + (0.898 + 0.438i)T \)
17 \( 1 + (0.776 - 0.629i)T \)
19 \( 1 + (-0.843 + 0.537i)T \)
23 \( 1 + (0.965 - 0.261i)T \)
29 \( 1 + (-0.965 - 0.261i)T \)
31 \( 1 + (0.455 - 0.890i)T \)
37 \( 1 + (0.726 + 0.686i)T \)
41 \( 1 + (-0.584 - 0.811i)T \)
43 \( 1 + (0.752 + 0.658i)T \)
47 \( 1 + (-0.169 + 0.985i)T \)
53 \( 1 + (0.999 - 0.0378i)T \)
59 \( 1 + (0.776 + 0.629i)T \)
61 \( 1 + (0.644 + 0.764i)T \)
67 \( 1 + (-0.822 - 0.569i)T \)
71 \( 1 + (0.521 + 0.853i)T \)
73 \( 1 + (0.881 - 0.472i)T \)
79 \( 1 + (-0.0944 - 0.995i)T \)
83 \( 1 + (0.614 - 0.788i)T \)
89 \( 1 + (-0.553 - 0.832i)T \)
97 \( 1 + (-0.455 - 0.890i)T \)
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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

−18.291817956769002614906997359258, −17.55514185517634067521939144090, −16.95579599456379782094704183252, −16.46376274143113546791084138797, −15.47179324597465097793410021451, −15.01539748026560607740489898601, −14.43784210739791340560433730665, −13.38898708031143099023710058905, −12.87607541672641676866329894758, −12.226969152915093847788251332674, −11.3740827515567956491204983029, −10.88077513668922173978915274997, −10.169345302452083354980823740738, −9.18509462110693631641230602063, −8.49638777959370097786373015160, −7.91225308758502892257955195065, −7.21463072335338265889742855460, −6.55783072933541614058927474752, −5.32942116122440677242345266194, −4.87705785513233217334629547826, −3.83537257990862445705546883772, −3.70289981945176018707148722542, −2.27137387532355906581027073716, −1.27611461933218518198697873923, −0.78793457610981384454155583763, 0.4895642052792901678743512427, 1.35777402517626436605885182283, 2.43846377885074885651927937483, 3.16833913083738644806474339727, 3.95434390163907645154169727712, 4.62363465536901645623221308005, 5.74284497826868233361344006594, 6.15563434986309019366140078392, 7.13337807735597235711635385711, 7.87339111790606441775522794991, 8.516632208727124544980038385589, 9.02087582337516190980145499775, 10.05561375421724621470439432565, 11.01431537515815079417108712710, 11.34617414317054750341558652186, 11.87515393494573821628240590038, 12.7245814903140742439389665718, 13.61875165638012772324554593898, 14.31245550492348685736277572205, 14.93732536527509913997991130569, 15.39006596577321218892859083725, 16.38368150291490122763696574993, 16.66559004745104953987035420744, 17.72548491772009795627547576967, 18.55013819607457284057946072449

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