Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.850 + 0.526i$
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 + (0.819 − 0.573i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.984 + 0.173i)23-s + (−0.642 − 0.766i)25-s + (0.573 − 0.819i)29-s + (0.766 + 0.642i)31-s + (−0.965 − 0.258i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.766 + 0.642i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯
L(s,χ)  = 1  + (−0.422 + 0.906i)5-s + (0.906 − 0.422i)11-s + (0.819 − 0.573i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.984 + 0.173i)23-s + (−0.642 − 0.766i)25-s + (0.573 − 0.819i)29-s + (0.766 + 0.642i)31-s + (−0.965 − 0.258i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.766 + 0.642i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.850 + 0.526i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.850 + 0.526i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.850 + 0.526i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (2797, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.850 + 0.526i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.849729275 + 0.5266568164i$
$L(\frac12,\chi)$  $\approx$  $1.849729275 + 0.5266568164i$
$L(\chi,1)$  $\approx$  1.126343382 + 0.1565358807i
$L(1,\chi)$  $\approx$  1.126343382 + 0.1565358807i

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

−17.53572759688438159759227768055, −16.95990281032559223432617167433, −16.38316073789732226492249277516, −15.59209379877787500185264473250, −15.30151716968319655634936571036, −14.30259547522408358141517703059, −13.593760245938646270089268410171, −13.13785406415160834928745736774, −12.28375912043785076292559675455, −11.74238524919432943611667441196, −11.21104310162551425116564085339, −10.41101652941873203535819228556, −9.38603731714171127566547002775, −8.92785790227401987422280740463, −8.56789875141132809235119648992, −7.52328192848382493998248960482, −6.85268329874027284267259734487, −6.29230885414754589754440087715, −5.21434160134108486297999357320, −4.6809601403252251131633057266, −4.02062245281801530411641512847, −3.32788956861334471869480986581, −2.236888870882213582921177897586, −1.38608110863600769294839597004, −0.708016623766501089678093328933, 0.77130287016911235098517531874, 1.598757551842527750108026549580, 2.73428359773218631265444901730, 3.267715089335186096855652539937, 3.94599752415582789184023433287, 4.69792344969284610325710329402, 5.74678537501504375936944252758, 6.38719375585445322516055005874, 6.85733626616277447478647305191, 7.75425547873938257567094017562, 8.338835729711893784002710744138, 9.075206584215544389345691186455, 9.84420269707718814561029607635, 10.62946577946905703056238829298, 11.17565112479110834736000915097, 11.65802309166523382709611694522, 12.43410195888529268952219819207, 13.26582907698895570298231789880, 13.98613774690388576948571849774, 14.3737685233129473435174568914, 15.27735691796338251092540951992, 15.69452635359486568766311616432, 16.310208225369283985278547439985, 17.26450758235069928261188732743, 17.77375073572235249139122985456

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