Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.157 + 0.987i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (0.0871 − 0.996i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (0.996 − 0.0871i)29-s + (−0.766 + 0.642i)31-s + (0.707 + 0.707i)37-s + (0.642 + 0.766i)41-s + (−0.906 − 0.422i)43-s + (−0.766 − 0.642i)47-s + (0.258 − 0.965i)53-s + i·55-s + ⋯
L(s,χ)  = 1  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (0.0871 − 0.996i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (0.996 − 0.0871i)29-s + (−0.766 + 0.642i)31-s + (0.707 + 0.707i)37-s + (0.642 + 0.766i)41-s + (−0.906 − 0.422i)43-s + (−0.766 − 0.642i)47-s + (0.258 − 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.157 + 0.987i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.157 + 0.987i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.157 + 0.987i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (347, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.157 + 0.987i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6113069144 + 0.5216645969i$
$L(\frac12,\chi)$  $\approx$  $0.6113069144 + 0.5216645969i$
$L(\chi,1)$  $\approx$  0.8175053061 + 0.004086609175i
$L(1,\chi)$  $\approx$  0.8175053061 + 0.004086609175i

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

−17.664017445901428387207929345, −16.742763473900870848536553289773, −16.12600960003883536268707670539, −15.72478865775620240976730162481, −14.96489102035745782769172691041, −14.31254116915522963100023042405, −13.764735305719178331111106153157, −12.75853508862560045532922842098, −12.28396496347428022583726387874, −11.567133587554715539230189830236, −11.14888388031692063213730330360, −10.29218988435554862801601268711, −9.43071208833009389506483211826, −9.01804512544515090825519948301, −8.05530657185019576513400686764, −7.44020539499509734098369962342, −6.919286341541011297456499349422, −6.23031578312928114665803090760, −5.07740177601947591589083548143, −4.49640791317126635743538093735, −4.02921426666205181119686702968, −2.991211467537573213728919283, −2.36089341512441998534314076639, −1.36418912270681471372115065400, −0.26196482325655695333063712372, 0.83208537682915759889050327118, 1.60718483536888711158471322138, 2.99003136241462572104514584585, 3.29212714593120690351513868225, 4.06640640878757103531109450571, 4.94365966761985640848890209324, 5.62888123458898183594955223209, 6.34988466969457426710638430235, 7.20151849840273413085990352459, 8.04770656354058093556208643411, 8.23480628840631193447611218839, 9.07475167391426689291706156201, 9.98586598122362692223572757267, 10.68051515280189200702959343637, 11.26580497225411627548133204073, 11.892448911438587367616190223500, 12.5141177884453789541433649068, 13.3445934372385552061146454623, 13.75314360398482512402117916411, 14.87463319276938407680010570978, 15.132648672171325138138715986493, 16.070602947319808903007369140683, 16.26924570788330767862697541495, 17.14583488664741664692805801938, 18.04548148058642986768508989585

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