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Properties

Label 1-6048-6048.781-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.0311 - 0.999i$
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  + (0.996 − 0.0871i)5-s + (−0.0871 + 0.996i)11-s + (−0.906 − 0.422i)13-s − 17-s + (0.707 + 0.707i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (0.422 + 0.906i)29-s + (0.173 − 0.984i)31-s + (−0.965 − 0.258i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.173 − 0.984i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.996 − 0.0871i)5-s + (−0.0871 + 0.996i)11-s + (−0.906 − 0.422i)13-s − 17-s + (0.707 + 0.707i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (0.422 + 0.906i)29-s + (0.173 − 0.984i)31-s + (−0.965 − 0.258i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.173 − 0.984i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.0311 - 0.999i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.0311 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9789016919 - 0.9488613707i\)
\(L(\frac12)\) \(\approx\) \(0.9789016919 - 0.9488613707i\)
\(L(1)\) \(\approx\) \(1.085522858 - 0.09115397869i\)
\(L(1)\) \(\approx\) \(1.085522858 - 0.09115397869i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.996 - 0.0871i)T \)
11 \( 1 + (-0.0871 + 0.996i)T \)
13 \( 1 + (-0.906 - 0.422i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.422 + 0.906i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.819 - 0.573i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.422 - 0.906i)T \)
89 \( 1 - iT \)
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

−17.73200084002334420626463938239, −17.368469300142300806205420072639, −16.599062745021721980084829206458, −15.927504541040707443090177937223, −15.30207632230027553253952437351, −14.41384796471488787764693131652, −13.85925299747399356093231816975, −13.4191576726600560071802948068, −12.78704318143142795701390932000, −11.702244575957464035017987115826, −11.42593837791375616572462729168, −10.4623709195493582591742961296, −9.887883160607690426865294336378, −9.25363144923096455481288927364, −8.65018161954311556855735505503, −7.85150017375128789781079913091, −6.864133604885672005461833258459, −6.52281398694617459814318705271, −5.58750092976628931032220018864, −5.07077632328604305235168826196, −4.30359323995182863805089018378, −3.187154889167602566041082849018, −2.67287677326790301046377495998, −1.83093294711710634118560162622, −1.01372456763587554152070661246, 0.33264735676076227992023355005, 1.63298420969729522972575366692, 2.14849035560744039069207276215, 2.83155358883482551025598591036, 3.872807204601440322791125834284, 4.76042606113287078553455621372, 5.24086963740290798973511099616, 5.98595104417062376726870813402, 6.895390271832800518658392945277, 7.23370966634850680988879341482, 8.29396015630165204234338243055, 8.9171195027762839845528881969, 9.696527132214450965602625143023, 10.22245598753020632734637591349, 10.63323196241784497475668020157, 11.80616196897072325000201305534, 12.33502192674529372994080491163, 12.91381001716871290249312566933, 13.648744008706205102775091936652, 14.239420837177983283869789558546, 14.91039851197399999434343291222, 15.482320666874358964267773872181, 16.33606568748160269767563029099, 17.06922948280408019126741072474, 17.45444209080460994169096930574

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