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Properties

Label 1-6008-6008.5-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.691 - 0.722i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (−0.993 + 0.117i)3-s + (0.0711 − 0.997i)5-s + (0.577 − 0.816i)7-s + (0.972 − 0.232i)9-s + (0.570 + 0.821i)11-s + (0.994 + 0.108i)13-s + (0.0460 + 0.998i)15-s + (0.630 − 0.775i)17-s + (−0.999 + 0.0167i)19-s + (−0.478 + 0.878i)21-s + (0.906 + 0.421i)23-s + (−0.989 − 0.141i)25-s + (−0.938 + 0.344i)27-s + (−0.129 − 0.991i)29-s + (0.521 + 0.853i)31-s + ⋯
L(s)  = 1  + (−0.993 + 0.117i)3-s + (0.0711 − 0.997i)5-s + (0.577 − 0.816i)7-s + (0.972 − 0.232i)9-s + (0.570 + 0.821i)11-s + (0.994 + 0.108i)13-s + (0.0460 + 0.998i)15-s + (0.630 − 0.775i)17-s + (−0.999 + 0.0167i)19-s + (−0.478 + 0.878i)21-s + (0.906 + 0.421i)23-s + (−0.989 − 0.141i)25-s + (−0.938 + 0.344i)27-s + (−0.129 − 0.991i)29-s + (0.521 + 0.853i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.512772413 - 0.6460793178i\)
\(L(\frac12)\) \(\approx\) \(1.512772413 - 0.6460793178i\)
\(L(1)\) \(\approx\) \(0.9664047611 - 0.2015034617i\)
\(L(1)\) \(\approx\) \(0.9664047611 - 0.2015034617i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.993 + 0.117i)T \)
5 \( 1 + (0.0711 - 0.997i)T \)
7 \( 1 + (0.577 - 0.816i)T \)
11 \( 1 + (0.570 + 0.821i)T \)
13 \( 1 + (0.994 + 0.108i)T \)
17 \( 1 + (0.630 - 0.775i)T \)
19 \( 1 + (-0.999 + 0.0167i)T \)
23 \( 1 + (0.906 + 0.421i)T \)
29 \( 1 + (-0.129 - 0.991i)T \)
31 \( 1 + (0.521 + 0.853i)T \)
37 \( 1 + (-0.884 + 0.467i)T \)
41 \( 1 + (-0.425 + 0.904i)T \)
43 \( 1 + (-0.693 + 0.720i)T \)
47 \( 1 + (0.926 + 0.375i)T \)
53 \( 1 + (-0.728 + 0.684i)T \)
59 \( 1 + (0.880 - 0.474i)T \)
61 \( 1 + (0.957 + 0.289i)T \)
67 \( 1 + (0.903 - 0.429i)T \)
71 \( 1 + (0.448 - 0.893i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.999 + 0.0167i)T \)
83 \( 1 + (0.999 + 0.0418i)T \)
89 \( 1 + (0.276 - 0.960i)T \)
97 \( 1 + (0.418 - 0.908i)T \)
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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

−17.69686242813265859301301532094, −17.29297633989270928948736273591, −16.58414109346607223297122827154, −15.83130398375027313071530732349, −15.148411060120394581195041840642, −14.6742301100172994482133949150, −13.88774724914270515729182395751, −13.13538120597469369407243181836, −12.39592740092240922673575604138, −11.73860106668688827255452824757, −11.12986895797306137338226304056, −10.70497342599044718342817554233, −10.165502531981810822473104537902, −9.00117916566381679574824112415, −8.492022371434259128296080216684, −7.691048117636129362764433558237, −6.62142179381058642748512284675, −6.4644956964764937725109726232, −5.58674644838592584899639056726, −5.194281368825130140740717577184, −3.9141753685939896884497890022, −3.557854504909421271251776558782, −2.36891546272424551592857774994, −1.70061630108108335655291097411, −0.7655674392818986933733165830, 0.71675181659466837963916943136, 1.28333913829601293674189836087, 1.90605598004997631534383179656, 3.43378411202468526135131039514, 4.17191943834523238012159417455, 4.75723018692305301278077721946, 5.16799289138501013581593802452, 6.16206559905991267588401351425, 6.739423338383325857246442688596, 7.49335942267302534698298004739, 8.25260803894283410961023002595, 9.01296794189637581912493981704, 9.81723249748267326152040265392, 10.27557000093545345166861112275, 11.2446655125254091640106782845, 11.56407983934670816878997978885, 12.343025943807818087964186835775, 12.94053675852459822008353803266, 13.57894753617108314160082706314, 14.263998188622266226197634704705, 15.25987764201009230103477219152, 15.72055588773004287874590502343, 16.64434494556335116057594576840, 16.892635274246091360783022162167, 17.51150126496587867874199787164

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