Properties

Degree 1
Conductor $ 2^{3} \cdot 751 $
Sign $0.691 - 0.722i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $0.691 - 0.722i$
motivic weight  =  \(0\)
character  :  $\chi_{6008} (5, \cdot )$
Sato-Tate  :  $\mu(750)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6008,\ (0:\ ),\ 0.691 - 0.722i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.512772413 - 0.6460793178i$
$L(\frac12,\chi)$  $\approx$  $1.512772413 - 0.6460793178i$
$L(\chi,1)$  $\approx$  0.9664047611 - 0.2015034617i
$L(1,\chi)$  $\approx$  0.9664047611 - 0.2015034617i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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