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Properties

Label 1-4008-4008.83-r1-0-0
Degree $1$
Conductor $4008$
Sign $-0.986 - 0.162i$
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  + (−0.0567 + 0.998i)5-s + (0.914 + 0.404i)7-s + (0.0189 − 0.999i)11-s + (0.421 + 0.906i)13-s + (0.132 + 0.991i)17-s + (0.988 + 0.150i)19-s + (−0.614 − 0.788i)23-s + (−0.993 − 0.113i)25-s + (0.614 − 0.788i)29-s + (0.387 − 0.922i)31-s + (−0.455 + 0.890i)35-s + (−0.316 + 0.948i)37-s + (−0.700 − 0.713i)41-s + (−0.974 − 0.225i)43-s + (0.351 + 0.936i)47-s + ⋯
L(s)  = 1  + (−0.0567 + 0.998i)5-s + (0.914 + 0.404i)7-s + (0.0189 − 0.999i)11-s + (0.421 + 0.906i)13-s + (0.132 + 0.991i)17-s + (0.988 + 0.150i)19-s + (−0.614 − 0.788i)23-s + (−0.993 − 0.113i)25-s + (0.614 − 0.788i)29-s + (0.387 − 0.922i)31-s + (−0.455 + 0.890i)35-s + (−0.316 + 0.948i)37-s + (−0.700 − 0.713i)41-s + (−0.974 − 0.225i)43-s + (0.351 + 0.936i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.986 - 0.162i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ -0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07228333416 + 0.8833681667i\)
\(L(\frac12)\) \(\approx\) \(-0.07228333416 + 0.8833681667i\)
\(L(1)\) \(\approx\) \(1.047773022 + 0.2896201655i\)
\(L(1)\) \(\approx\) \(1.047773022 + 0.2896201655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.0567 + 0.998i)T \)
7 \( 1 + (0.914 + 0.404i)T \)
11 \( 1 + (0.0189 - 0.999i)T \)
13 \( 1 + (0.421 + 0.906i)T \)
17 \( 1 + (0.132 + 0.991i)T \)
19 \( 1 + (0.988 + 0.150i)T \)
23 \( 1 + (-0.614 - 0.788i)T \)
29 \( 1 + (0.614 - 0.788i)T \)
31 \( 1 + (0.387 - 0.922i)T \)
37 \( 1 + (-0.316 + 0.948i)T \)
41 \( 1 + (-0.700 - 0.713i)T \)
43 \( 1 + (-0.974 - 0.225i)T \)
47 \( 1 + (0.351 + 0.936i)T \)
53 \( 1 + (0.0944 + 0.995i)T \)
59 \( 1 + (0.132 - 0.991i)T \)
61 \( 1 + (-0.822 - 0.569i)T \)
67 \( 1 + (-0.0567 - 0.998i)T \)
71 \( 1 + (-0.553 - 0.832i)T \)
73 \( 1 + (0.942 + 0.334i)T \)
79 \( 1 + (-0.521 + 0.853i)T \)
83 \( 1 + (-0.644 - 0.764i)T \)
89 \( 1 + (-0.776 + 0.629i)T \)
97 \( 1 + (-0.387 - 0.922i)T \)
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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

−17.92292329809372002273781503363, −17.46190282855073245630025869530, −16.51272282061042261707957791520, −15.9748537374342287255361470412, −15.32107389880355189733622516205, −14.53980264329687517739946034497, −13.7026488629795312609519356772, −13.32316265395809436257729099889, −12.290480834606382141174745109154, −11.920870801471629889904661514936, −11.17754633147352007150280527164, −10.17421059445220660974694612130, −9.75441094529742261451519556546, −8.76325307852806417523839705273, −8.240403958572378260120219140709, −7.44465398050961137055293875429, −6.962821969180791639192962648178, −5.55594200380908286411901833621, −5.18761087913646741771376711399, −4.54203596741650853891682812503, −3.69832483286128245030380513166, −2.762021436234420968035950432795, −1.55145979795613078365915606189, −1.19124794197831825328031767909, −0.126538203906783870448639109226, 1.16722030592146023676374113527, 1.97533712433294613052838696136, 2.78330486906982881693543287347, 3.62893299990042026596509218796, 4.308458037097745432065502256347, 5.29322627946552458588064608417, 6.21280936343069456768689982197, 6.45007379501032672169454274856, 7.66400044216826277328898075204, 8.14215489376403135972907333155, 8.81137876134935317993815997327, 9.76810341963971737877268863037, 10.51630106265164700978638768427, 11.14268670072367294835761059636, 11.72024542275237545453725283230, 12.23090012029290896182007476521, 13.54570574882906425177430135404, 13.96630343506364355285989583243, 14.44544349847841378309420338116, 15.36008402663801358704553400987, 15.71253355344725710668227117285, 16.79204653921484222228255876563, 17.24803407387172742825581747956, 18.24895200876361486069332395775, 18.63578750357737171710203293033

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