Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.911 + 0.411i$
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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.911 + 0.411i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (277, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.911 + 0.411i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6853142840 + 0.1474537905i$
$L(\frac12,\chi)$  $\approx$  $0.6853142840 + 0.1474537905i$
$L(\chi,1)$  $\approx$  0.7010290560 + 0.09579216365i
$L(1,\chi)$  $\approx$  0.7010290560 + 0.09579216365i

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

−17.63092486914429631486469416390, −16.86486558238585518827903752453, −16.400663051564294058640407776761, −15.651223240140820895221939248729, −15.16775698734170180253337299832, −14.52958210409469782529638862917, −13.55751898797614293479025224592, −12.985803509539008326796601188929, −12.522289121410803408739924487741, −11.66574702832456756692217864484, −11.021450703055655342148542091230, −10.64735614250384880207115272274, −9.47475855831663103422243947996, −8.91837946369932684853262589845, −8.42550701234288458452724948374, −7.37264926984828176704874921090, −7.26400199215262707894779028458, −6.0839334403912681485627377336, −5.325265528693927333586805924592, −4.68229342856167408933071904596, −4.07003870985517129275420364932, −3.12341081372962081697020082602, −2.515154148600239726602626037750, −1.44279624612849254359594097601, −0.412027232221406097665075788, 0.39486815633996730431529528059, 1.941082438865783882897885471811, 2.4098585686396516286730401971, 3.25401908121543657239163928656, 4.20980272048864552696717997818, 4.59977546319341181659427098098, 5.46138678169553089651528548564, 6.51732745174155807867969319888, 6.99445222528291552010439384660, 7.64009071866176782727194554494, 8.28381686298281216383503770019, 9.056068400434894917815972595322, 9.901446537219524716485449550325, 10.58430305596088799428596458060, 10.982073837651713660107771318, 11.99305289467777329829770720696, 12.34588885158174346307385631945, 13.03900088156734881157416944028, 13.910613802823645305497103319881, 14.668455584783296100555246631110, 15.18333561632583534753664561681, 15.54367575303454614791101658045, 16.46537041232255658226516462186, 17.09176366733260216303593030912, 17.77900292289236509020476347841

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