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Properties

Label 1-6048-6048.3811-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.162 - 0.986i$
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  + (0.0871 − 0.996i)5-s + (−0.996 + 0.0871i)11-s + (0.996 + 0.0871i)13-s + (−0.5 + 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.342 + 0.939i)23-s + (−0.984 − 0.173i)25-s + (0.0871 + 0.996i)29-s + (0.766 − 0.642i)31-s + (−0.707 + 0.707i)37-s + (−0.642 − 0.766i)41-s + (0.422 − 0.906i)43-s + (−0.766 − 0.642i)47-s + (−0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.0871 − 0.996i)5-s + (−0.996 + 0.0871i)11-s + (0.996 + 0.0871i)13-s + (−0.5 + 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.342 + 0.939i)23-s + (−0.984 − 0.173i)25-s + (0.0871 + 0.996i)29-s + (0.766 − 0.642i)31-s + (−0.707 + 0.707i)37-s + (−0.642 − 0.766i)41-s + (0.422 − 0.906i)43-s + (−0.766 − 0.642i)47-s + (−0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9105608461 - 0.7729552651i\)
\(L(\frac12)\) \(\approx\) \(0.9105608461 - 0.7729552651i\)
\(L(1)\) \(\approx\) \(0.9391805321 - 0.1491896395i\)
\(L(1)\) \(\approx\) \(0.9391805321 - 0.1491896395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0871 - 0.996i)T \)
11 \( 1 + (-0.996 + 0.0871i)T \)
13 \( 1 + (0.996 + 0.0871i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.0871 + 0.996i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (0.422 - 0.906i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
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 \( 1 - iT \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.0871 - 0.996i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−17.91818849259041286850711342448, −17.45360248446833567573600260673, −16.37260661381870207273574218221, −15.68473426894912491250790815367, −15.46965321729141995042838545526, −14.52875136689295722639095856338, −13.847280468007745440602386121172, −13.41316489513457285254451071317, −12.704380705492564972815097060973, −11.76159515171196505946605758621, −11.073907311862395480751782039338, −10.72603770766090103116751032541, −9.98054419226252722205319644280, −9.26583990596146694093318193703, −8.36969099855535947997305023275, −7.861611213624839210529084770212, −6.96070778762228159609742386288, −6.45264053643685848141489178287, −5.80099909461898241353861944516, −4.84665671910288589552985469125, −4.234110984725672752095646702518, −3.127913489451098645175969435257, −2.750946383344681238597898552542, −1.991288366476132576710870354443, −0.768186376332895426453413917989, 0.36926632269919264724104195956, 1.680461207997415043973845902628, 1.82818994464493813420526948573, 3.234183529184429276399661798121, 3.81561199382172603000786884708, 4.64884230774618565848126615036, 5.34458131043799435051511565183, 5.94368467295306023322352496613, 6.661269986145769930416564363450, 7.72947199320862421922648183115, 8.27998296715518872660583156379, 8.71079075684997049318025302488, 9.58124995810178497343889181794, 10.29190016923753074245678315936, 10.850765436678748432778588473001, 11.7272048190134353927488570640, 12.37926830856651630145178552516, 13.00665586514276159316903138879, 13.500671459893826279880228230078, 14.11803926022745619440939437270, 15.136027043884015446524445977264, 15.77944882904068719004506489786, 16.04907775036103257541023217058, 17.08545859885455750422332528780, 17.32459606629544766193482115930

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