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Properties

Label 1-6048-6048.5227-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.935 - 0.352i$
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  + (0.819 − 0.573i)5-s + (0.573 − 0.819i)11-s + (−0.573 − 0.819i)13-s + (−0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (0.819 + 0.573i)29-s + (0.173 − 0.984i)31-s + (−0.707 − 0.707i)37-s + (0.984 + 0.173i)41-s + (−0.996 − 0.0871i)43-s + (−0.173 − 0.984i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.819 − 0.573i)5-s + (0.573 − 0.819i)11-s + (−0.573 − 0.819i)13-s + (−0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (0.819 + 0.573i)29-s + (0.173 − 0.984i)31-s + (−0.707 − 0.707i)37-s + (0.984 + 0.173i)41-s + (−0.996 − 0.0871i)43-s + (−0.173 − 0.984i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2540564541 - 1.394219437i\)
\(L(\frac12)\) \(\approx\) \(0.2540564541 - 1.394219437i\)
\(L(1)\) \(\approx\) \(1.029162622 - 0.4120529311i\)
\(L(1)\) \(\approx\) \(1.029162622 - 0.4120529311i\)

Euler product

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

−17.91824877303727300396210830041, −17.33935122572306736754143057785, −16.8728278336833870925095384372, −16.084973091320321288096081031455, −15.215894652313876862450176058678, −14.5801230546225649669410341261, −14.202747508786438975440108231707, −13.55617229126815127300833046174, −12.558274280940742822940895410183, −12.24376563567157966611981016437, −11.3894050145746982619610459256, −10.50863465691995206016785071965, −10.09392804133226726796502903238, −9.47483452115777899838472089779, −8.727015133826181032377723782324, −7.96249254326508220460025436531, −7.072620744839839847448302983083, −6.38701255266086254401724853911, −6.16471877560303683375763662743, −4.95877469747974323949596681767, −4.40707725537786183493983572504, −3.59326194839473067088930647988, −2.63556597863169574072219376493, −1.89967561953454615175611031807, −1.450262128686998953201775475253, 0.33846168154618588545944769872, 1.123615238255142990765550410229, 2.129632600231982219194708153212, 2.76377596325748277160366386372, 3.62519685724230161017732558744, 4.60767565631670466076876731266, 5.16912922066335449521109289356, 5.856138398226579591330885655360, 6.528623379974265787687604876844, 7.29023116918846067950448708143, 8.16093729572247694501716197918, 8.842494469990340747247418287351, 9.384863304562846258932536578015, 10.01024757855150725579416804835, 10.780381148128896714699317516613, 11.51557787087540384942113471514, 12.1523662563107041280558953440, 12.942499877300820512908079184439, 13.51134033234392216702928164163, 13.98830360414963179963714816956, 14.72380336644013233212350364570, 15.577651844819747433604679370624, 16.12782429392721367768372884120, 16.815112630821863822556574369596, 17.54585446612999953425951412214

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