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Properties

Label 1-6048-6048.1685-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: $\chi_{6048} (1685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01091993602 - 0.05992678717i\)
\(L(\frac12)\) \(\approx\) \(0.01091993602 - 0.05992678717i\)
\(L(1)\) \(\approx\) \(0.7893262339 + 0.05112876321i\)
\(L(1)\) \(\approx\) \(0.7893262339 + 0.05112876321i\)

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

−18.014298930759226093750053459005, −17.15184658634722734368669375048, −16.68650326834214351873976830417, −16.108683148647737450493286576383, −15.34922991005971820498746078170, −14.815957520925811277055977165362, −14.10999892456184469453553750522, −13.40552695444251101291114848657, −12.51338940718246859186833023552, −12.09504064169127170990282112066, −11.54605149021549314872419058859, −10.88996258819546367577072842920, −9.76257162414124481739371018114, −9.41961545869388511637426045027, −8.70647574323269860540117977874, −7.91781832929282104502581788387, −7.11137499966992656725283261689, −6.89224923770325635242671665252, −5.692766133845447019940618650538, −4.87208222068983170423011058617, −4.42007163798480745287246005810, −3.73456197920873053250285855622, −2.77789712996023273482848883916, −1.937506324025649617228514970, −1.04189337533397751006954185282, 0.01757110504408810613751521726, 1.14899906029989104190561361895, 2.08155976162533281509536241528, 3.10280514858032726217929753449, 3.69526448990811014476860462434, 4.07324781112237647319335458421, 5.50482224940944593276889013156, 5.6588039940615016459776538781, 6.76978276709246582447071776137, 7.3141355318174454229548987637, 8.18540623648742211080448196819, 8.396692597920935114170561848184, 9.5507680839319401693526297686, 10.212686155438875041215000034803, 10.781722223079312488749641863211, 11.51909636085567050480574635172, 12.12795339866512381024280260638, 12.62020404304520655612955204586, 13.58698680968030611193644622041, 14.324507238710880878583876799960, 14.70768007501690174338669968436, 15.49956333492246017322058545086, 15.9911612607748781758481315691, 16.814841836910504440662910953406, 17.30589486556367075494478162960

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