Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.0943 + 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0943 + 0.995i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0943 + 0.995i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.0943 + 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (293, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.0943 + 0.995i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4170819656 + 0.3794045140i$
$L(\frac12,\chi)$  $\approx$  $0.4170819656 + 0.3794045140i$
$L(\chi,1)$  $\approx$  0.8004666831 - 0.09266090483i
$L(1,\chi)$  $\approx$  0.8004666831 - 0.09266090483i

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

−17.95626717797690206787550201374, −16.804341804364802084225476059135, −16.13600008965381844113620493825, −15.53528193420384756875962347320, −14.99494333763655504251120090604, −14.32216903906034530183892131117, −13.72063989691198277521618301985, −12.93572771226641900148988691614, −12.078935404185681954783637142250, −11.83258272189646391002114486115, −10.72470610271872721069586083507, −10.17787680918502210631986127737, −10.018093163020278273216017380568, −8.7154162172891193643864427670, −7.98767517741505249390914267229, −7.59442252487351172546015599776, −6.84095102974044643608085979611, −6.06336526827784875018217606233, −5.33859588810218116339138568215, −4.59879968633744921368211528210, −3.68940286267240112039873748995, −2.960575782161453070148076334719, −2.49627215669995367291134426142, −1.40514180479540162395712104792, −0.17413045500890258781124630429, 0.815033695850242581200285023606, 1.79404121387664829710334289494, 2.53464318880559072019566030266, 3.544015634325567574972706552810, 4.30700105282892080068250986983, 4.76033754372446567949012307431, 5.74523907346472815309629866611, 6.17494544768007151303695077373, 7.33749918845388382787954061844, 7.8017384234577713718569942982, 8.573630393974040288601049588544, 9.06031627765905460196160778428, 9.855187487061692877873922167747, 10.61686454417306838955206738125, 11.33921367866517812678517310053, 11.95002738709947161100811724839, 12.646571422128476046890418169, 13.16685271000616503812752402722, 13.8874081910558178300464336722, 14.53397624492070823260372838952, 15.496115175731925433462699713588, 15.89076565771089786054048906125, 16.46692322219506807616192423717, 17.18882537044392314595331787753, 17.69753906883036472080434249933

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