Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.735 + 0.677i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.573 + 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.0871 − 0.996i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.996 − 0.0871i)29-s + (−0.939 + 0.342i)31-s + (−0.965 − 0.258i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (0.939 + 0.342i)47-s + (0.965 + 0.258i)53-s i·55-s + ⋯
L(s,χ)  = 1  + (0.573 + 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.0871 − 0.996i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.996 − 0.0871i)29-s + (−0.939 + 0.342i)31-s + (−0.965 − 0.258i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (0.939 + 0.342i)47-s + (0.965 + 0.258i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.735 + 0.677i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.735 + 0.677i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.735 + 0.677i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (115, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.735 + 0.677i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.223000333 + 0.8684760262i$
$L(\frac12,\chi)$  $\approx$  $2.223000333 + 0.8684760262i$
$L(\chi,1)$  $\approx$  1.309159614 + 0.2250469494i
$L(1,\chi)$  $\approx$  1.309159614 + 0.2250469494i

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

−17.494106253808322668347120276377, −16.74151140633301053566011079048, −16.623018676085819007794641450895, −15.82689750397975883872888258606, −14.8758243318210759042196437311, −14.32417324403259133399323349837, −13.55483756133237214860255886921, −13.2499227380609768197080449429, −12.27395773778251280693377898557, −11.71293545505264005061324451888, −11.21123806679288950210383906108, −10.17315703720225307341434424718, −9.48363267573578900299755232250, −9.04277482193576552547657915424, −8.5077136310618672645274740947, −7.41607281909140031480749099358, −6.954062820779472937959802746633, −5.92915843561309313088324377656, −5.477008236268220837116634403634, −4.76128061304016071383458097351, −3.865911872347799781553636262191, −3.27566569342902119591801500739, −2.13523794485515286177293737821, −1.45555712916386082904747002062, −0.740405398368982830488479680526, 0.88923151462604235331274600354, 1.757339393760673584455944097493, 2.5010207021623917519861979390, 3.45311085423845422673958134514, 3.76503972557805082064987643758, 5.10443270865241841093099962024, 5.54394339285260939246973640707, 6.28968602033922847487402485859, 7.1315314346909743152550953082, 7.50423080757516540853115741268, 8.413589751326824064699710339280, 9.37498331208357336658371322211, 9.75042087849451580081349695006, 10.57702201544367484584600088789, 10.94285420559417496449903019217, 12.00365309588748335204044543897, 12.43584403594154273208738438879, 13.20121628958769057045500934233, 14.02906909393599473593018489159, 14.55541755427174536557843309031, 14.95341583973233990388966716409, 15.73461605692058323989178885738, 16.68370853092444871984709094726, 17.09224460786538006169662385320, 17.85085816898224414058446000516

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