Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.411 + 0.911i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.411 + 0.911i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.411 + 0.911i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.411 + 0.911i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (2893, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.411 + 0.911i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.140334946 + 1.382379615i$
$L(\frac12,\chi)$  $\approx$  $2.140334946 + 1.382379615i$
$L(\chi,1)$  $\approx$  1.353084547 + 0.3268361491i
$L(1,\chi)$  $\approx$  1.353084547 + 0.3268361491i

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

−17.663310778909443312630403044749, −16.923919056098851742889977999173, −16.13455057022322227780775842135, −15.97632015415502764932496863375, −14.87636833847047586962768248750, −14.119451672323537190037130828568, −13.72961732062540444657329150750, −12.92461511532758695163909366839, −12.60092349791507412852158786005, −11.49088015878558244664016773254, −10.95080338080644270084529695196, −10.45572866089604228235950109810, −9.21567736610233611561926230577, −9.0457126571141179472062304780, −8.54495854029431974951878505639, −7.432632900477649222029277548687, −6.70684052974805488147057381014, −5.99597616924758084797243629924, −5.488792042485394780312213688433, −4.63593047832096016880078672295, −3.921054287193996396922490295880, −3.025982740116855760051317366025, −2.26869571723012579141000228, −1.29309352823666740952271971739, −0.72800398099981499221377595042, 1.045457471942046094896780554037, 1.82829034824072926856705669253, 2.41752151671460407271246141858, 3.465348198471370083867754684433, 3.9912276428881519963699828446, 4.97906507672517945752565203883, 5.73157952915820154833415444753, 6.364822347210206596151884796205, 6.9656097707325807326336782755, 7.65282436492106074252303696426, 8.57935165209798792182948916893, 9.37308313953008047194367256930, 9.7213237271515645490209426874, 10.52972631422346527823905461124, 11.29267676604055977519053653524, 11.66696917034261918772173878474, 12.78617224931361443821956960067, 13.23390971044892113666111894545, 13.94037357043886492144451122103, 14.447395753712919878172182690141, 15.30047728827845584120155609803, 15.64371528735548633674761421240, 16.7222364284773725257576526945, 17.17653547045203722853561990750, 17.92266869103797984758271353735

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