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Properties

Label 1-6048-6048.445-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.852 + 0.523i$
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 + (0.422 + 0.906i)13-s + (0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.984 + 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.906 − 0.422i)29-s + (−0.939 − 0.342i)31-s + (−0.707 + 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.573 − 0.819i)43-s + (0.939 − 0.342i)47-s + (−0.965 − 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.422 + 0.906i)13-s + (0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.984 + 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.906 − 0.422i)29-s + (−0.939 − 0.342i)31-s + (−0.707 + 0.707i)37-s + (0.342 − 0.939i)41-s + (−0.573 − 0.819i)43-s + (0.939 − 0.342i)47-s + (−0.965 − 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.852 + 0.523i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.852 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1778382139 + 0.6295614484i\)
\(L(\frac12)\) \(\approx\) \(0.1778382139 + 0.6295614484i\)
\(L(1)\) \(\approx\) \(1.012702006 + 0.08398543476i\)
\(L(1)\) \(\approx\) \(1.012702006 + 0.08398543476i\)

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 \( 1 + (0.422 + 0.906i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.906 - 0.422i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.573 - 0.819i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (-0.0871 + 0.996i)T \)
61 \( 1 + (-0.906 - 0.422i)T \)
67 \( 1 + (-0.573 + 0.819i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.906 + 0.422i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−17.59590008408093655325434213281, −16.75505725863591396304020169444, −16.22117204186810159948918132375, −15.41235268659354531812096676072, −14.76527887879124356277096358460, −14.155319103100140176599081344511, −13.44595336178824957630542238645, −12.96765341432258002525105321277, −12.34858600843662307754296219766, −11.21212506887405275494574945481, −10.73832893265332580375076414369, −10.34413305814700773376496948431, −9.39611716476428999695156053784, −8.85183172063476278614842129038, −8.01355493172594234831716157612, −7.441817291343439271710315238082, −6.402985204938955439294698053732, −5.92188160918478331627284856431, −5.42855308352642019844538668741, −4.51021211253315172385597759151, −3.3442620727478741267894972440, −3.10155714940016646998985767482, −2.029844944571261574322398294690, −1.3744940028401099851051347000, −0.14475981033980457295545526402, 1.313806283907664778482675382106, 1.91833705832630419792209903037, 2.50576628363872557591443422557, 3.71313591946903250756130022017, 4.27639487866618389096114431794, 5.24344163453239402350761616691, 5.65539110556874742288023921942, 6.46527658304083947737694168956, 7.246659499058640466834676051623, 7.87773625299934993015589218238, 8.823132028604176000639666876564, 9.33176681667831895665359239078, 10.03564195082680133625088188017, 10.446786180115286281974003869812, 11.500217654720646060185026741427, 12.130977315511791602082325555095, 12.6896202652503598553937512785, 13.482742599635613382601262708111, 13.99522627739671556500805958551, 14.54456485885229301819489020936, 15.425817349578600092290871127639, 16.09855842375587135370460096164, 16.76433502776841896332026052276, 17.19039448868726602622460297953, 18.069072974318328381152272486721

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