Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.0322 + 0.999i$
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 + (−0.819 + 0.573i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.573 + 0.819i)29-s + (−0.173 + 0.984i)31-s + (0.707 − 0.707i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (0.258 + 0.965i)53-s i·55-s + ⋯
L(s,χ)  = 1  + (−0.573 − 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.819 + 0.573i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.573 + 0.819i)29-s + (−0.173 + 0.984i)31-s + (0.707 − 0.707i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (0.258 + 0.965i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0322 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0322 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.0322 + 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (173, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.0322 + 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8581029271 + 0.8308977540i$
$L(\frac12,\chi)$  $\approx$  $0.8581029271 + 0.8308977540i$
$L(\chi,1)$  $\approx$  0.9561581779 + 0.06015405879i
$L(1,\chi)$  $\approx$  0.9561581779 + 0.06015405879i

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

−17.46981556893802449831237758014, −16.98074709940107692404138473856, −16.1171840173466483020658337986, −15.60961723921002559715856087324, −14.8091735832747284910607138814, −14.407777445738754525958807476730, −13.73007187635925802277797866317, −12.97105110458892101924756596094, −12.081000524220863222278898088607, −11.43938896256001800764074874545, −11.26923625273074852446349167658, −10.10794984305732818118393246125, −9.68623102816167565532251620124, −8.967915280155747755409970975233, −7.844384053339359821417647087773, −7.587304899559432826693683901430, −6.89288087796068159965027634926, −6.003919296891151520458204042691, −5.42500800833816875489411222178, −4.48195094794091846119331890943, −3.67205255994852585804988590262, −3.083633685280340989081290455992, −2.45631037583242260652362022971, −1.26475386555532435784518815193, −0.34180167801973182944278891966, 1.06599077783811207067250143341, 1.56047592659707528223479884815, 2.660014517661825605845552652904, 3.54404502578398070264666634292, 4.29514103456911707770879854345, 4.8027590150883855137499041297, 5.55225004328472304095555285625, 6.44899556264353190468943771652, 7.31937207925934998745967236003, 7.630226105712742206607034180560, 8.70428432231196201488195055598, 9.13912436463282239940353417930, 9.72079798685039664653048242916, 10.65097416882900223660475704387, 11.35326401457454684038516954953, 12.226942775250640889733578619540, 12.333731654240069761790591792373, 13.13037819349745611833929505788, 14.01491166064999518233222537069, 14.79765578070715915846964102397, 15.01787277516048187811785675777, 16.12831522722247616867417217937, 16.57560069693415176792610720247, 17.00543306971674599045724396202, 17.828427326930427479006424136007

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