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Properties

Label 1-6048-6048.5-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.954 + 0.297i$
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  + (−0.422 − 0.906i)5-s + (−0.906 − 0.422i)11-s + (−0.819 − 0.573i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.984 + 0.173i)23-s + (−0.642 + 0.766i)25-s + (−0.573 − 0.819i)29-s + (−0.766 + 0.642i)31-s + (−0.965 + 0.258i)37-s + (0.984 − 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)5-s + (−0.906 − 0.422i)11-s + (−0.819 − 0.573i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.984 + 0.173i)23-s + (−0.642 + 0.766i)25-s + (−0.573 − 0.819i)29-s + (−0.766 + 0.642i)31-s + (−0.965 + 0.258i)37-s + (0.984 − 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.954 + 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3732876510 + 0.05685792693i\)
\(L(\frac12)\) \(\approx\) \(0.3732876510 + 0.05685792693i\)
\(L(1)\) \(\approx\) \(0.6389856407 - 0.1469539052i\)
\(L(1)\) \(\approx\) \(0.6389856407 - 0.1469539052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.422 - 0.906i)T \)
11 \( 1 + (-0.906 - 0.422i)T \)
13 \( 1 + (-0.819 - 0.573i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.573 - 0.819i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.0871 - 0.996i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (-0.573 + 0.819i)T \)
61 \( 1 + (0.996 - 0.0871i)T \)
67 \( 1 + (0.906 - 0.422i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.573 - 0.819i)T \)
89 \( 1 + iT \)
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

−17.86512159494082893851147582751, −17.11263925396239441513746265905, −16.17658336801233928643542614280, −15.70524275426105056145278267163, −15.02725040527737926475555985858, −14.47479706924830401734612783284, −13.90295531152636716257278635468, −12.83417144901156605027843579358, −12.644185928485003397482270768881, −11.49318727895987415743358314778, −11.13345887760850190549811218445, −10.479393725525712126147943165358, −9.73118557178071078479316425425, −9.11009847841148644872912513089, −8.11876666996357305627873776709, −7.609123115071256349317633428261, −6.85096412958319670982191740201, −6.43498785334365579529154634077, −5.402753163454800631596392289993, −4.606418390817044576477306191633, −4.0477051913761345095616091204, −3.10283294843914176794272748353, −2.31423154260848575626405420076, −1.917543003589448022517472169888, −0.18207281904851484295476488695, 0.44080706535153653171638431651, 1.75658049702404925059177995775, 2.315404159021613510550397387910, 3.39697757473876496276334816884, 4.06100805983636923021052763137, 4.847467842097874674187085921660, 5.44039584916062495941403123796, 6.093541197895798344511635237452, 7.12696864144900200440625763889, 7.82832528204782025227563179127, 8.31427621627851589915305071736, 8.967318820991421603815395351993, 9.77920806017006871757270281459, 10.461720397122849761886050953778, 11.11113388257021297862775816173, 11.94228553116509423512031095682, 12.52468981399247490370148983165, 13.03619284095184160108450352968, 13.685458284948789747074058553505, 14.48369401415468083567861920896, 15.38280389276108420278284477359, 15.67351889890234575558996267719, 16.438367793741807013095084456, 17.021197785244720834712432656291, 17.68831091685101106591306884463

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