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Properties

Label 1-6048-6048.5045-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.677 - 0.735i$
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 + (0.0871 − 0.996i)13-s − 17-s + (0.707 − 0.707i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (−0.996 + 0.0871i)29-s + (0.939 + 0.342i)31-s + (0.965 − 0.258i)37-s + (−0.642 − 0.766i)41-s + (−0.906 − 0.422i)43-s + (0.939 − 0.342i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.573 − 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.0871 − 0.996i)13-s − 17-s + (0.707 − 0.707i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (−0.996 + 0.0871i)29-s + (0.939 + 0.342i)31-s + (0.965 − 0.258i)37-s + (−0.642 − 0.766i)41-s + (−0.906 − 0.422i)43-s + (0.939 − 0.342i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5219106583 - 1.191174240i\)
\(L(\frac12)\) \(\approx\) \(0.5219106583 - 1.191174240i\)
\(L(1)\) \(\approx\) \(1.000944579 - 0.2920949430i\)
\(L(1)\) \(\approx\) \(1.000944579 - 0.2920949430i\)

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 \( 1 + (0.0871 - 0.996i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.996 + 0.0871i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.906 - 0.422i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (-0.996 - 0.0871i)T \)
61 \( 1 + (0.422 - 0.906i)T \)
67 \( 1 + (0.819 + 0.573i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.996 + 0.0871i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.21521626204040190114655346795, −17.19981670262142150293732658038, −16.716020706434807686432262563071, −16.00250995472695465245742231873, −15.23162665904437473339290906205, −14.69296933044542926723008871402, −13.93445021254537781530461042736, −13.425343361266283564695921279944, −12.93826327180308110695324785936, −11.77697261980931360346446883877, −11.353173236471319285676017955924, −10.64829626814337746572167022131, −10.05932646450186494248731707035, −9.33915654588663921190914210868, −8.67603416392032336790836962384, −7.82691552982991932294669226115, −7.152899110232069354774596167325, −6.37720755206699278903399895903, −5.95951431996946308692656377804, −5.0447033835298968692327274955, −4.30152462787512289290445217715, −3.38027944091408597786911002723, −2.66887338366642080612033605874, −2.0734366008275333986801107597, −1.10502170001701835586413672995, 0.33392570295689394969529685812, 1.24351812606816091200129078876, 2.17327220679295357439402372692, 2.7766732973396708936877905186, 3.75592595348326766539976385967, 4.68730770199500525277194975858, 5.26381204557136203300320800099, 5.67939970469583548354416383125, 6.744224843394308718398910450200, 7.38475689381062649401650846686, 8.16949820219660976134087553610, 8.80839257774639378545377203794, 9.50296277541670799991505802681, 10.0748479219693944931443031496, 10.80073177436587017104651024100, 11.52179757445241630384629890468, 12.34554948960522137261459120826, 13.04758525966106648208771432624, 13.335035454540365807554987982733, 13.97346934240102558321119606944, 15.12692599638369844384951501175, 15.48089541655129887572487481511, 16.02278067805400787326160983673, 17.0774937952159407573519660120, 17.32753864508242719492515884319

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