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Properties

Label 1-6048-6048.2371-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.0902 - 0.995i$
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  + (−0.906 + 0.422i)5-s + (0.422 − 0.906i)11-s + (−0.573 + 0.819i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (0.819 − 0.573i)29-s + (0.766 − 0.642i)31-s + (−0.258 − 0.965i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.766 − 0.642i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.906 + 0.422i)5-s + (0.422 − 0.906i)11-s + (−0.573 + 0.819i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.984 − 0.173i)23-s + (0.642 − 0.766i)25-s + (0.819 − 0.573i)29-s + (0.766 − 0.642i)31-s + (−0.258 − 0.965i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.766 − 0.642i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.0902 - 0.995i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.0902 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7110275164 - 0.7783870095i\)
\(L(\frac12)\) \(\approx\) \(0.7110275164 - 0.7783870095i\)
\(L(1)\) \(\approx\) \(0.8875951204 - 0.08133915379i\)
\(L(1)\) \(\approx\) \(0.8875951204 - 0.08133915379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.906 + 0.422i)T \)
11 \( 1 + (0.422 - 0.906i)T \)
13 \( 1 + (-0.573 + 0.819i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.819 - 0.573i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.996 + 0.0871i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.819 - 0.573i)T \)
61 \( 1 + (-0.0871 - 0.996i)T \)
67 \( 1 + (0.422 + 0.906i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.819 + 0.573i)T \)
89 \( 1 - iT \)
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

−17.77709970382665600426389872372, −17.14074926351222882115074425697, −16.62423606560801106027009600472, −15.90466977832375930321018241236, −15.1342592086001913320804846869, −14.80991233003291792560402356945, −14.106681277041635366448158449594, −12.99002604803558608408996566121, −12.583948070802001736518710153543, −12.034301938531312915543270986489, −11.460429823037208299275286314721, −10.42675556978497773060538809654, −10.06052305312343674921433462556, −9.18064626878079214378372691264, −8.389620247980138235075828832557, −7.90186214571622551683158324069, −7.18470061254676036474968608231, −6.5626111222467678547641087231, −5.53291496787597193525035455669, −4.81067529498627254401397687463, −4.36356711602264476181150800644, −3.331658158490387763444563824476, −2.89723420724890023716529850470, −1.60698542529071089030316160280, −0.98550830669409123873474403022, 0.32537307423113190682974759119, 1.202180303980939659079886262822, 2.43318074358306024607608529031, 3.00283108498709268805575839116, 3.84558633972408876935332996284, 4.428337468250304811176534471541, 5.20957106555149480758011996688, 6.21931431572604580837046058710, 6.75310553416443849523552027633, 7.42910185134227121383440364402, 8.19363575984676602451840692021, 8.74848571987812745753491337278, 9.53697612703632925513728425205, 10.29642104672275777114927674295, 11.160646571064571199731356983675, 11.443452409353629299253280224119, 12.21007523764262684458017803340, 12.79330214129834079876432991245, 13.80275455071785709976154143387, 14.26397988247206796405222524857, 14.93352779995153917533413718945, 15.52531647724728015761070426815, 16.25224404695895738981798058577, 16.86817565410331279368726633282, 17.33770309231839703338595742100

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