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  :  $\chi_{48} (29, \cdot )$
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

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

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