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Properties

Label 1-6048-6048.3229-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.629 + 0.776i$
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  + (0.573 + 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.819 + 0.573i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.642 + 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.573 + 0.819i)29-s + (0.173 − 0.984i)31-s + (−0.707 + 0.707i)37-s + (0.984 + 0.173i)41-s + (0.0871 − 0.996i)43-s + (−0.173 − 0.984i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.573 + 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.819 + 0.573i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.642 + 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.573 + 0.819i)29-s + (0.173 − 0.984i)31-s + (−0.707 + 0.707i)37-s + (0.984 + 0.173i)41-s + (0.0871 − 0.996i)43-s + (−0.173 − 0.984i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.629 + 0.776i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.629 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7891080699 + 1.656176961i\)
\(L(\frac12)\) \(\approx\) \(0.7891080699 + 1.656176961i\)
\(L(1)\) \(\approx\) \(1.104773467 + 0.4331605600i\)
\(L(1)\) \(\approx\) \(1.104773467 + 0.4331605600i\)

Euler product

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

−17.56450780864709389928113533463, −16.68439005852599983888473502287, −16.30508037988073258368382271704, −15.718640821677384810450625004700, −14.61758664426041049863230882230, −14.17460092760724786250234286163, −13.65172180095117974996595352709, −12.78017879645470428833434530321, −12.198960386485603290142148477933, −11.749295088428188977305517935485, −10.8068201959499476821013505467, −10.00572038092676174634687984269, −9.43932446476299929921848900080, −8.983592151573027247954622204624, −8.01826613749534083495324290867, −7.57140539490632852485084650099, −6.53935979436258779542828829325, −5.89374645415570650347612838304, −5.21969774153231430779136766427, −4.645245187743203634241219968356, −3.71967475493986795392541347310, −2.92438462672795880496677577751, −2.0950205896546238699730653577, −1.18362815863635044363734012367, −0.46867953461074142223315499025, 1.250516250951550065915084630169, 1.9145654910863439986496469005, 2.626161578730137335711114701220, 3.586339451660843654873393065544, 4.08992072982769392518025800603, 5.18553475070194467946203216710, 5.74301652121952602558190551373, 6.55991635722005537003449721316, 7.18429933400605236941746490885, 7.64177449896117918914007645809, 8.726463921976853469871417896427, 9.53639138803146188570597452409, 9.84675595789804766152388429133, 10.56052335716442564871180406022, 11.45795753082534426685230267297, 11.901641361733382557813618128312, 12.63672296451285294536634729839, 13.520081091326986694396766747934, 14.06901330702560628232913671045, 14.68009119643288461031074539127, 15.10909673689247407422307353536, 15.94602788624750915045941942692, 16.86560616279354191694505727396, 17.287259860826089467546837770825, 17.81606512327701200464717335629

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