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Properties

Label 1-6048-6048.4723-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.849 + 0.527i$
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  + (0.422 − 0.906i)5-s + (−0.906 + 0.422i)11-s + (0.906 + 0.422i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (0.422 + 0.906i)29-s + (−0.939 + 0.342i)31-s + (0.707 − 0.707i)37-s + (0.342 + 0.939i)41-s + (0.819 + 0.573i)43-s + (0.939 + 0.342i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.422 − 0.906i)5-s + (−0.906 + 0.422i)11-s + (0.906 + 0.422i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (0.422 + 0.906i)29-s + (−0.939 + 0.342i)31-s + (0.707 − 0.707i)37-s + (0.342 + 0.939i)41-s + (0.819 + 0.573i)43-s + (0.939 + 0.342i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.849 + 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009383840033 + 0.03289254647i\)
\(L(\frac12)\) \(\approx\) \(0.009383840033 + 0.03289254647i\)
\(L(1)\) \(\approx\) \(0.8925744371 - 0.1236488960i\)
\(L(1)\) \(\approx\) \(0.8925744371 - 0.1236488960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.422 - 0.906i)T \)
11 \( 1 + (-0.906 + 0.422i)T \)
13 \( 1 + (0.906 + 0.422i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.422 + 0.906i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (0.819 + 0.573i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.258 - 0.965i)T \)
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 \( 1 - iT \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.422 - 0.906i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−17.59849965032423807844617550024, −16.89202815235464254575305823094, −15.939279100840031658540361699147, −15.461479372446773230307649830823, −14.91005695524269398743135996924, −14.101080218300015580813564640792, −13.428241286000610445493814593330, −13.0860607147095244139945935944, −12.18791505790253548567237105397, −11.25614301566318674551684439070, −10.69906355683862181561027838716, −10.40016198797827328421428700687, −9.53989905899465157600584657153, −8.625842950460498698239044558560, −8.11829300579112165436712228479, −7.3447323088743541329286664742, −6.53901668752928066396145952049, −5.82822203109856550426782543963, −5.57743990370578606345615896615, −4.13224287157963975890998166116, −3.84711391958579486121720578311, −2.625249353789652738029936731942, −2.382172423913408067169937091690, −1.30375152589213107758609785162, −0.008391596824808501847964848354, 1.1170711039676064121207327557, 1.97566097458456051563244347466, 2.54381387983571667269807746724, 3.68545257419406222777891955294, 4.44992028089009115230888568637, 4.96776376305341433640583967412, 5.82051609742692503916712636140, 6.35857720734667570339481724238, 7.28234506460804394320398000867, 8.0597776356469518505981777784, 8.65835427867995603844621944683, 9.29595999838917007052133035389, 9.918609996920524116935494598621, 10.76179868269595652248238705371, 11.26247832952747281796684087015, 12.27621800608129252767306147081, 12.75016772256460676536148838064, 13.26831328345592188669951174820, 14.01920684778971048502702344953, 14.570219867195628795138841543276, 15.58290104488873643810242082793, 16.17829204934843910255344152023, 16.403592333077666897731812255411, 17.39176064361061234124278653254, 18.09951877050900257410349850733

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