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Properties

Label 1-6048-6048.5819-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.0311 - 0.999i$
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  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (−0.906 + 0.422i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.422 + 0.906i)29-s + (−0.173 − 0.984i)31-s + (−0.965 + 0.258i)37-s + (0.342 − 0.939i)41-s + (0.819 − 0.573i)43-s + (−0.173 + 0.984i)47-s + (−0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (−0.906 + 0.422i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.422 + 0.906i)29-s + (−0.173 − 0.984i)31-s + (−0.965 + 0.258i)37-s + (0.342 − 0.939i)41-s + (0.819 − 0.573i)43-s + (−0.173 + 0.984i)47-s + (−0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.0311 - 0.999i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.0311 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4660629337 - 0.4808181768i\)
\(L(\frac12)\) \(\approx\) \(0.4660629337 - 0.4808181768i\)
\(L(1)\) \(\approx\) \(0.7590523547 - 0.04036656288i\)
\(L(1)\) \(\approx\) \(0.7590523547 - 0.04036656288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.996 - 0.0871i)T \)
11 \( 1 + (-0.0871 - 0.996i)T \)
13 \( 1 + (-0.906 + 0.422i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (-0.422 + 0.906i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (0.819 - 0.573i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
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 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.422 + 0.906i)T \)
89 \( 1 - iT \)
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

−17.72743408360012652004721194197, −17.33220280227688128223528088189, −16.44089190184405172034216664400, −15.94456137551406847417385017554, −15.054689173121977578292516064, −14.83585968888482820538504156877, −14.14469095323187312724792778929, −13.05672881212783443476244264764, −12.45175512223742513356319503570, −12.13257059678821440713755370189, −11.30743266540233951124391043593, −10.54569445254983199526772892409, −10.00506267145848127214221988980, −9.24627062446563834746400582877, −8.352044143765554589925637581383, −7.81523403532623558504185379250, −7.17094009728148870506972016794, −6.618749121920456728796561771419, −5.573240378277626878998147665827, −4.743236622254632586829090511, −4.34818881957903018659341980727, −3.41034488508981378494579223578, −2.68706549904325937335860078434, −1.91267766772456585565513873774, −0.72159086402562839449238474759, 0.23961022960069973216140264035, 1.29897925935228103931826860533, 2.22886789609968486773920087962, 3.28612458849486762897599709783, 3.67630739555324733004554024334, 4.48859187851977142401453172768, 5.3451794714859381128514029659, 5.92745959980566861544805199430, 6.88886540365960455513573990249, 7.6248354658209233869338041018, 8.00320801439529409251172851498, 8.841828921809682321895901946615, 9.47884458034458227063601626766, 10.33531943205346048704993760951, 11.03125625083417124090847593069, 11.57452003048201650733122300951, 12.3969999907231925569025078623, 12.61582223393947806566637404886, 13.76413629735200262164898485603, 14.314941491907183000209640584676, 14.86913424186500818390763962616, 15.72638769596450986511844678672, 16.13130647722766438439990881704, 16.97944625491045973802618941150, 17.20026295408986687417268674422

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