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Properties

Label 1-6048-6048.4979-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.776 + 0.629i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.776 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.724967324 + 0.6116528284i\)
\(L(\frac12)\) \(\approx\) \(1.724967324 + 0.6116528284i\)
\(L(1)\) \(\approx\) \(1.137005508 + 0.1335176328i\)
\(L(1)\) \(\approx\) \(1.137005508 + 0.1335176328i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.573 + 0.819i)T \)
11 \( 1 + (-0.819 - 0.573i)T \)
13 \( 1 + (0.819 - 0.573i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.573 + 0.819i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (0.0871 - 0.996i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.422 + 0.906i)T \)
61 \( 1 + (0.573 - 0.819i)T \)
67 \( 1 + (0.0871 + 0.996i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.573 + 0.819i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.73905197113674861063544163535, −16.805669894250898921458255488493, −16.483804832052518634784539773880, −15.69319434396731902857826688097, −15.131671824063953878800135883801, −14.25794477310757428068272919103, −13.550419080297596328283749609132, −13.0835565249897642975002569368, −12.53296917689377886174550235835, −11.69639372052211275714011825415, −11.07663865011197287534381077616, −10.16250411365720927641352077989, −9.684114740740315217216507079673, −9.052188479084093821870707944072, −8.04480014875757696404641878309, −7.9675535785695650564754779508, −6.657103053287805482543850672406, −6.111542408489089892070187912538, −5.45567134511579896020638744045, −4.595523129926275051803066185501, −4.14113657032958515822089824321, −3.11939971742215061832426788650, −2.06474848361477867547186323996, −1.695398720972599868875085108045, −0.57948164751501962814100992162, 0.76680550468800371380967054889, 1.778774118672200649593117067134, 2.61669148990545798420484816381, 3.25382386270466385888713408201, 3.825271599480392383925699403010, 5.24535677029493009883461257921, 5.418937938788845469225987954136, 6.250177360687865599693568199679, 7.130230741201182682245261604814, 7.513895982203835492026277635793, 8.50774373506271993111899861334, 9.10340648332820948620434412318, 9.95440050144123448803681587686, 10.47551022413771829154370793804, 11.16653519918949614787960815748, 11.60242434789797058323406505182, 12.65478227495289962549022823433, 13.39218817583349776008534235690, 13.79304196764032445896029857024, 14.320786914892843960512931887024, 15.35446021361350215974869923111, 15.71015496512494222386990429160, 16.34652538639850294755728124380, 17.23865900962941017605285072357, 18.07812240255338123596411658151

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