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Properties

Label 1-6048-6048.1949-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: $\chi_{6048} (1949, \cdot )$
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

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

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