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Properties

Label 1-4008-4008.707-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.0884 + 0.996i$
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  + (0.387 + 0.922i)5-s + (−0.974 − 0.225i)7-s + (−0.132 − 0.991i)11-s + (−0.0944 − 0.995i)13-s + (−0.800 + 0.599i)17-s + (0.489 − 0.872i)19-s + (−0.997 + 0.0756i)23-s + (−0.700 + 0.713i)25-s + (0.997 + 0.0756i)29-s + (−0.351 + 0.936i)31-s + (−0.169 − 0.985i)35-s + (0.776 − 0.629i)37-s + (−0.752 − 0.658i)41-s + (0.0189 + 0.999i)43-s + (−0.584 − 0.811i)47-s + ⋯
L(s)  = 1  + (0.387 + 0.922i)5-s + (−0.974 − 0.225i)7-s + (−0.132 − 0.991i)11-s + (−0.0944 − 0.995i)13-s + (−0.800 + 0.599i)17-s + (0.489 − 0.872i)19-s + (−0.997 + 0.0756i)23-s + (−0.700 + 0.713i)25-s + (0.997 + 0.0756i)29-s + (−0.351 + 0.936i)31-s + (−0.169 − 0.985i)35-s + (0.776 − 0.629i)37-s + (−0.752 − 0.658i)41-s + (0.0189 + 0.999i)43-s + (−0.584 − 0.811i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0884 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0884 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.0884 + 0.996i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.0884 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5282077397 + 0.4833658647i\)
\(L(\frac12)\) \(\approx\) \(0.5282077397 + 0.4833658647i\)
\(L(1)\) \(\approx\) \(0.8450003732 + 0.004148304149i\)
\(L(1)\) \(\approx\) \(0.8450003732 + 0.004148304149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.387 + 0.922i)T \)
7 \( 1 + (-0.974 - 0.225i)T \)
11 \( 1 + (-0.132 - 0.991i)T \)
13 \( 1 + (-0.0944 - 0.995i)T \)
17 \( 1 + (-0.800 + 0.599i)T \)
19 \( 1 + (0.489 - 0.872i)T \)
23 \( 1 + (-0.997 + 0.0756i)T \)
29 \( 1 + (0.997 + 0.0756i)T \)
31 \( 1 + (-0.351 + 0.936i)T \)
37 \( 1 + (0.776 - 0.629i)T \)
41 \( 1 + (-0.752 - 0.658i)T \)
43 \( 1 + (0.0189 + 0.999i)T \)
47 \( 1 + (-0.584 - 0.811i)T \)
53 \( 1 + (-0.614 + 0.788i)T \)
59 \( 1 + (-0.800 - 0.599i)T \)
61 \( 1 + (0.455 - 0.890i)T \)
67 \( 1 + (0.387 - 0.922i)T \)
71 \( 1 + (-0.822 + 0.569i)T \)
73 \( 1 + (-0.726 - 0.686i)T \)
79 \( 1 + (-0.644 - 0.764i)T \)
83 \( 1 + (-0.982 - 0.188i)T \)
89 \( 1 + (-0.0567 + 0.998i)T \)
97 \( 1 + (0.351 + 0.936i)T \)
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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

−18.20562501628535040378397176238, −17.389971088552732814075192512652, −16.73804537586954626265109360537, −16.06036650612671924104652721249, −15.73494370789061088030592444509, −14.70706804515460799233729675828, −13.94918762003806210084562715371, −13.27077472277203394962327762, −12.747245736016424068619974034717, −11.935019253349179452979634588716, −11.62391793816267068703261697881, −10.1536119176083868081783754569, −9.84644397706599965885517424141, −9.23638186038301156175001467682, −8.52580254293949514542747708672, −7.66419318041074957732997781705, −6.79132419556434122958798857969, −6.18550096683948110708384519794, −5.415206970707048817153038472688, −4.48288275358829789874126319664, −4.097138327847839666392829935089, −2.85801429565226284999971838162, −2.0994565763605353612318043674, −1.354839332013282283451146652716, −0.1609091126167800578943217807, 0.552183576804354025826688517639, 1.771968212123186833743580064610, 2.90214820491808962697553451352, 3.10403307722676595033141948079, 4.00663532097651700191700115164, 5.12327074031706886772456578724, 5.99589048294199504448312957444, 6.4000313042091478099965862241, 7.14753973321590843707900333996, 7.96546469555753207655168833296, 8.77132608490121709418150819514, 9.58814810666446618732775264876, 10.27989867773709944081752457818, 10.771098994993010347271437479914, 11.42252116007369832654739203721, 12.43801175652092649925168715410, 13.1167327364947584351487153942, 13.702117339167375130062654378809, 14.24260190579930458518678068032, 15.17916675611941387953720242490, 15.77807873042542870446647772156, 16.26447491699965250501276684601, 17.29429347014269486996485244184, 17.807018428033838366345073284882, 18.40719004366187502106415908296

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