Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.677 - 0.735i$
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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.677 - 0.735i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (3139, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.677 - 0.735i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.280315160 - 0.5609675777i$
$L(\frac12,\chi)$  $\approx$  $1.280315160 - 0.5609675777i$
$L(\chi,1)$  $\approx$  0.9333975546 - 0.1640125581i
$L(1,\chi)$  $\approx$  0.9333975546 - 0.1640125581i

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

−17.824294796705855155513009605429, −17.26657231372669908818521289922, −16.34831080154635186298446884484, −15.74519679663950126365137120797, −15.12795202054435787553261863360, −14.659218301419649556880108335115, −14.01002944984132126547292342405, −13.00413742034417708628437161469, −12.60487071096701471459356810216, −11.89414348068621994901105133465, −11.00183162511088630119692803186, −10.59282420589215916780367874968, −9.973333385323993752542122044765, −9.23529372682607302796083695861, −8.08993344589296838109421864260, −7.76288679680935243081482501877, −7.263104049467832150452071982794, −6.20447430282633833124382066794, −5.72038560733304726063161842763, −4.80444766154036607219874040800, −4.05165026332461841131992849131, −3.170342668388195502974781439858, −2.77310248013326793619397807365, −1.77634500515546961435114009410, −0.66487284759709543833229724888, 0.55988704642047674211708776893, 1.30212678380614720710217750199, 2.40434952430071151221997396558, 3.09357383553775336982146393521, 4.09255126439647089396213934345, 4.56299586934920129245127013654, 5.350279307264179075796182567955, 6.01614244114386704634409748095, 6.961214226237028668148794988539, 7.59615317084875641058413912288, 8.34652680872588999198619360938, 8.89022166419982742321218856428, 9.443093664018001483090430615549, 10.504340238141260433390145803353, 10.99098589497777533841483822022, 11.706565452794548908978754758457, 12.52475953686788504876113570639, 12.79691967991326493062259006840, 13.71793807925831892952029986956, 14.27430324418487454862724310770, 15.10176103373013070065542722899, 15.77631265325609527962472284805, 16.38645531135057161106272266339, 16.74176434686025252429663651301, 17.525800508776421853554299287959

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