Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-0.317 - 0.948i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.134 − 0.990i)2-s + (−0.963 + 0.266i)4-s + (0.753 − 0.657i)5-s + (0.393 + 0.919i)8-s + (−0.753 − 0.657i)10-s + (0.473 − 0.880i)11-s + (0.134 + 0.990i)13-s + (0.858 − 0.512i)16-s + (0.963 + 0.266i)17-s + (0.309 + 0.951i)19-s + (−0.550 + 0.834i)20-s + (−0.936 − 0.351i)22-s + (0.550 + 0.834i)23-s + (0.134 − 0.990i)25-s + (0.963 − 0.266i)26-s + ⋯
L(s,χ)  = 1  + (−0.134 − 0.990i)2-s + (−0.963 + 0.266i)4-s + (0.753 − 0.657i)5-s + (0.393 + 0.919i)8-s + (−0.753 − 0.657i)10-s + (0.473 − 0.880i)11-s + (0.134 + 0.990i)13-s + (0.858 − 0.512i)16-s + (0.963 + 0.266i)17-s + (0.309 + 0.951i)19-s + (−0.550 + 0.834i)20-s + (−0.936 − 0.351i)22-s + (0.550 + 0.834i)23-s + (0.134 − 0.990i)25-s + (0.963 − 0.266i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.317 - 0.948i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.317 - 0.948i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-0.317 - 0.948i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (209, \cdot )$
Sato-Tate  :  $\mu(70)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ -0.317 - 0.948i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.174504427 - 1.632175317i$
$L(\frac12,\chi)$  $\approx$  $1.174504427 - 1.632175317i$
$L(\chi,1)$  $\approx$  0.9696796697 - 0.6414986975i
$L(1,\chi)$  $\approx$  0.9696796697 - 0.6414986975i

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

−17.95932101912500942655311913504, −17.14068043083443220218276093794, −16.837222442721122296205450477049, −15.79348948418705607206950203256, −15.30072502677447001630014097874, −14.61298042935776833863568860063, −14.23389536659146974329723558822, −13.4654757186130784805037013339, −12.83548911192268997482081460133, −12.20444289662407409417070559254, −11.10378532156747701828865388701, −10.35871506566345809080472806270, −9.86408564247846973578149927617, −9.27425098571014919430868695872, −8.494474081148110712241029512384, −7.723678768052685897967630694464, −6.994936739957720967774009296625, −6.58836176466112219228784498912, −5.83720815650021677924457117934, −5.007542429201844474959802475254, −4.66049014860075554286867940605, −3.30755827658708126474282222532, −2.93642085223990299509776861682, −1.63883853129361679362096399985, −0.89404482555085235240369605019, 0.675727714951647921929163928309, 1.45096194372706132711562106560, 1.94901858757698482350594821155, 2.97212489408284812978940879181, 3.73731809099412403407647725340, 4.30385130556780478778991878350, 5.30777588162956317752875477422, 5.74780175217362300544411363285, 6.569154273586287941814174078460, 7.75152621365367511157912704537, 8.37222294039464846524606858926, 8.98080515409293495167224941384, 9.77747065513114098160310206138, 9.92310864991518236217842406314, 11.02115359015417924696706826667, 11.6150119989297521769989437460, 12.12602846546485972745413384578, 12.88959186908031501116032064748, 13.50984446378234172353051930096, 14.11069348619174043655251869076, 14.44378445316589822852082136801, 15.68315526466650494008741259130, 16.53030534209033763978501992008, 17.03926601360867021122173214159, 17.33575119532479356483349036920

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