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Properties

Label 1-6048-6048.3733-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  + (−0.0871 − 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

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

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: $\chi_{6048} (3733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.852 - 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2497765928 - 0.8842290419i\)
\(L(\frac12)\) \(\approx\) \(0.2497765928 - 0.8842290419i\)
\(L(1)\) \(\approx\) \(0.9201486875 - 0.2363924685i\)
\(L(1)\) \(\approx\) \(0.9201486875 - 0.2363924685i\)

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

−18.09567103264447801201341550309, −17.2631118485486977714964818234, −16.69864749415036359417187825301, −16.10686715179915960337693755901, −15.13547283865538586835656577092, −14.555498949490232235879427255655, −14.34792137195974089082369343744, −13.52230487517486795794769251001, −12.55978332906136116464744747572, −11.99030345816582845123397422876, −11.43850456062011730209797026082, −10.687512007815207851407361238850, −9.91708821831544131123392834144, −9.59256288923557327649969601590, −8.5634620221316102254727108288, −7.85948526646018287616403479424, −7.14665564284282590382836542084, −6.5606175116605921238999396092, −5.99175566577719346991990128791, −4.953281811939731786269277067411, −4.35006953311000794997325747139, −3.33280074110496648639129642582, −2.93029594713049302577016644453, −2.01231801570735889661672188085, −1.10478153720839275764912446756, 0.23852786959591837338344281028, 1.30009152695316701956098030644, 1.88935733954822467516391891059, 2.9302025679767660890393328987, 3.84748635300589693284469406511, 4.457568773923820416920185496999, 5.115937722594760571064788664044, 5.84920577534321501733650218199, 6.651911525141649907301427641297, 7.4107751803853688785719103061, 8.08526839639811889342375036792, 8.87777729966967690547979578738, 9.36113912582251141701405522066, 9.965991526480744381585779645571, 10.8772491301514938924741644051, 11.65151196155156364903083342727, 12.277644157817764005751604716148, 12.62216594434659652221764121524, 13.57213697903726328792977985990, 14.040519175507311809299620827502, 14.95501574672349388464420971322, 15.42411187797128071274190230114, 16.17612708328141548564145840886, 16.95300710123026630954215216192, 17.39063681725321394583610090374

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