Properties

Degree 1
Conductor $ 2^{4} \cdot 3 $
Sign $-0.382 - 0.923i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·5-s − 7-s i·11-s i·13-s − 17-s i·19-s + 23-s − 25-s + i·29-s + 31-s + i·35-s + i·37-s + 41-s + i·43-s − 47-s + ⋯
L(s,χ)  = 1  i·5-s − 7-s i·11-s i·13-s − 17-s i·19-s + 23-s − 25-s + i·29-s + 31-s + i·35-s + i·37-s + 41-s + i·43-s − 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.382 - 0.923i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 48 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.382 - 0.923i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $-0.382 - 0.923i$
motivic weight  =  \(0\)
character  :  $\chi_{48} (5, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 48,\ (1:\ ),\ -0.382 - 0.923i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5798374110 - 0.8677880107i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5798374110 - 0.8677880107i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8378660543 - 0.3470554831i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8378660543 - 0.3470554831i\)

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

−33.85083039700226008940272714521, −33.1211035338160124079849133463, −31.56399845628199203736918143126, −30.65943687663393334575777186659, −29.37295541519771993617174780266, −28.46569938229418755244828126227, −26.81044197366514546407652563195, −26.039162895600592283254525317617, −24.91142508433301930640603332147, −23.13927562947809711985474278733, −22.53461145362158802335887369456, −21.171919455692234902885899324767, −19.5924705324034271458642320030, −18.69863706911562389238470524476, −17.35536937802984722652669141724, −15.86889955523466570947755183272, −14.72303261424883088358643763275, −13.37588819713542929158390236381, −11.923752431044553947411448863658, −10.44644123714894671910093945706, −9.3016485982255340114088614712, −7.26450304601908524448330798652, −6.27521599421773876975095638134, −4.081256636471305660353449939954, −2.41232286741208780390263096759, 0.60931577565694488504183911628, 3.1132652405032237626874346805, 4.96802771057281014616967016610, 6.46847296980554198635838731832, 8.3526374110098002213898527536, 9.45964687136241731017451506937, 11.08770029647687457830500388947, 12.743676746166520663159724581358, 13.45553208919998222255828314180, 15.4595680384709480072215824409, 16.395358624001599377904801609756, 17.617498510888476462334357497403, 19.28208216756627444202323153710, 20.15381907796775661448392044583, 21.53819236714168250557681280452, 22.73368821308457217874039635924, 24.10631331703809139504093821927, 25.04184519963900125302192934548, 26.34301077029950740350230430555, 27.62708801369315140819748479286, 28.78870540813313700656277115149, 29.611079601324156039358583399269, 31.22638330302583678552356606829, 32.33620722970762124753571310376, 32.8997089539539028467199552351

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