Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.811 + 0.583i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.573 + 0.819i)13-s + 17-s + (0.707 − 0.707i)19-s + (−0.984 − 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.819 − 0.573i)29-s + (−0.766 − 0.642i)31-s + (−0.258 + 0.965i)37-s + (0.984 + 0.173i)41-s + (−0.996 − 0.0871i)43-s + (−0.766 + 0.642i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯
L(s,χ)  = 1  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.573 + 0.819i)13-s + 17-s + (0.707 − 0.707i)19-s + (−0.984 − 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.819 − 0.573i)29-s + (−0.766 − 0.642i)31-s + (−0.258 + 0.965i)37-s + (0.984 + 0.173i)41-s + (−0.996 − 0.0871i)43-s + (−0.766 + 0.642i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.811 + 0.583i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.811 + 0.583i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.811 + 0.583i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (11, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ -0.811 + 0.583i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.01262841798 + 0.03919834855i$
$L(\frac12,\chi)$  $\approx$  $0.01262841798 + 0.03919834855i$
$L(\chi,1)$  $\approx$  0.7757387298 - 0.09896829270i
$L(1,\chi)$  $\approx$  0.7757387298 - 0.09896829270i

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

−17.66544819507593335414813990008, −16.53968311157736505086059826092, −16.0997413965999625019172914644, −15.56001924986504011682472963621, −14.67944495639761747812544862588, −14.48633406384071888802159844730, −13.48919570176923682598719685992, −12.56955398763658931474694680768, −12.3472284237641622457168952463, −11.474056504328888717695576012421, −10.848118754969326487738420008314, −10.13391308845077563669974797942, −9.66710969949845490261954465008, −8.55452494743707325218808468303, −7.96480177822497432252254100940, −7.43088945176822699695321348708, −6.87534060666379234304774597675, −5.72701577628479309544804903626, −5.35810760907143944965272656044, −4.323629796058544453744511067729, −3.547560082150380257205166218427, −3.19050163427112355617700557198, −2.0788521059505129777887350385, −1.24587773736553381965721487579, −0.0120559332483328627232480064, 0.97773917557450007347852342999, 1.80668342609042458783690581281, 2.99566251323859272705567313080, 3.5190638593413810560175892591, 4.26748622207526859823632009314, 4.99204092926772750264193653344, 5.80516662321008098902867165690, 6.41168191672813326003329893061, 7.53235118302592640770231102800, 7.80397000532028971326827304429, 8.60379254470318357735209437523, 9.22925188060358024424355791461, 9.938551470658000401977634582777, 10.918062346217726795990153855159, 11.44119244051574955329102112028, 11.87502720639309482082573319463, 12.720626961741846234610008712794, 13.375384071312411006554403674717, 14.01119340699735553859109587886, 14.67785461721652632922811395908, 15.61301119625340115285343495494, 15.92719136305962365011178225616, 16.660278241974016978981079760958, 17.005778235151382608101239247125, 18.22462528497777109759547914112

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