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Properties

Label 1-4008-4008.797-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.249 + 0.968i$
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  + (0.843 + 0.537i)5-s + (−0.521 − 0.853i)7-s + (−0.982 − 0.188i)11-s + (0.351 − 0.936i)13-s + (−0.243 + 0.969i)17-s + (−0.0567 − 0.998i)19-s + (−0.942 + 0.334i)23-s + (0.421 + 0.906i)25-s + (−0.942 − 0.334i)29-s + (0.672 + 0.739i)31-s + (0.0189 − 0.999i)35-s + (0.997 + 0.0756i)37-s + (−0.0944 + 0.995i)41-s + (−0.644 + 0.764i)43-s + (−0.898 + 0.438i)47-s + ⋯
L(s)  = 1  + (0.843 + 0.537i)5-s + (−0.521 − 0.853i)7-s + (−0.982 − 0.188i)11-s + (0.351 − 0.936i)13-s + (−0.243 + 0.969i)17-s + (−0.0567 − 0.998i)19-s + (−0.942 + 0.334i)23-s + (0.421 + 0.906i)25-s + (−0.942 − 0.334i)29-s + (0.672 + 0.739i)31-s + (0.0189 − 0.999i)35-s + (0.997 + 0.0756i)37-s + (−0.0944 + 0.995i)41-s + (−0.644 + 0.764i)43-s + (−0.898 + 0.438i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9266929998 + 0.7179270190i\)
\(L(\frac12)\) \(\approx\) \(0.9266929998 + 0.7179270190i\)
\(L(1)\) \(\approx\) \(0.9922437219 + 0.05978697696i\)
\(L(1)\) \(\approx\) \(0.9922437219 + 0.05978697696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.843 + 0.537i)T \)
7 \( 1 + (-0.521 - 0.853i)T \)
11 \( 1 + (-0.982 - 0.188i)T \)
13 \( 1 + (0.351 - 0.936i)T \)
17 \( 1 + (-0.243 + 0.969i)T \)
19 \( 1 + (-0.0567 - 0.998i)T \)
23 \( 1 + (-0.942 + 0.334i)T \)
29 \( 1 + (-0.942 - 0.334i)T \)
31 \( 1 + (0.672 + 0.739i)T \)
37 \( 1 + (0.997 + 0.0756i)T \)
41 \( 1 + (-0.0944 + 0.995i)T \)
43 \( 1 + (-0.644 + 0.764i)T \)
47 \( 1 + (-0.898 + 0.438i)T \)
53 \( 1 + (0.584 - 0.811i)T \)
59 \( 1 + (0.243 + 0.969i)T \)
61 \( 1 + (-0.974 + 0.225i)T \)
67 \( 1 + (-0.843 + 0.537i)T \)
71 \( 1 + (-0.914 - 0.404i)T \)
73 \( 1 + (0.965 + 0.261i)T \)
79 \( 1 + (0.700 - 0.713i)T \)
83 \( 1 + (0.752 - 0.658i)T \)
89 \( 1 + (-0.862 + 0.505i)T \)
97 \( 1 + (0.672 - 0.739i)T \)
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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

−18.467944800477860465037303427425, −17.85340573960693939383766541488, −16.649862556376714601345293469255, −16.516796320382022094931599097563, −15.686491693509563064512003007631, −15.0333122790568181936195805391, −14.03581730550052311128790955836, −13.60102290442215788397318486123, −12.8576920172259560965956114394, −12.24357708858879360842372678267, −11.62064480099552808798605940151, −10.64223863648995141367678091089, −9.81501310573788650345144155755, −9.450637643632101402204109175357, −8.66662571928803292911504039772, −8.01443561739072647521868983488, −7.02047915467473419485349701996, −6.16258996181420070034273869529, −5.69205451726441960467110116850, −4.95661114125777079410637995378, −4.13865410150034557579199856577, −3.08945810636373592071060774810, −2.17758357627673591393573745796, −1.82140973703782585003480467269, −0.350001266050531254836724453139, 0.92931988435099347426222588536, 1.92891248748492492513053683708, 2.87446494069732465720553497746, 3.35318090360373839721204985639, 4.38353349312153368710100886090, 5.24608998473035860129321965620, 6.13229434697781292574582097734, 6.44610365795367362643363530186, 7.52551158080002724735181882725, 7.99042540844014037333818354623, 8.97808784238813845787420950607, 9.88564455983668745341263162489, 10.308445276648845179886477151992, 10.83845879893746569100270266731, 11.58463587380837608371109427248, 12.87698074909549518000854709772, 13.24502334368810431581891299044, 13.55036356093528845570132360397, 14.60290424242025768736949200405, 15.1642421547578010031767000624, 15.928342172470084813685475850183, 16.62404121717155215388513602278, 17.37609517594137703063791346240, 18.006926247011922214179140975675, 18.3339342906973491166232071019

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