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Properties

Label 1-6048-6048.859-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: $\chi_{6048} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.954 - 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006170974633 + 0.04051411561i\)
\(L(\frac12)\) \(\approx\) \(0.006170974633 + 0.04051411561i\)
\(L(1)\) \(\approx\) \(0.8942207636 + 0.1000733660i\)
\(L(1)\) \(\approx\) \(0.8942207636 + 0.1000733660i\)

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

−17.36294201446668886713760285845, −16.65294918371882687058101386698, −16.103965674830585282933994198141, −15.54878806768476467096581472271, −14.65104076287697104689329462184, −13.87049412402678874463081074842, −13.62323729851425494519012854828, −12.45519616153690511117993139509, −12.26948854447553042036111861872, −11.68909666214756966532036890398, −10.398906702601341963850131468273, −10.01499270672120856128070825784, −9.54063486178288374454665177932, −8.55969760856419724192136097647, −8.028244561290064722463879530896, −7.37863767305351637417676111139, −6.50075335373002539173192197782, −5.60445845717313253020028018, −5.16244291864069577391958658929, −4.48445783229537657201816204014, −3.65028916085739361224552574805, −2.65232746093159382048889524922, −1.940664997456463001654725499900, −1.19291360214758668099046646323, −0.01005828347589514120492992817, 1.22394506490857632742845333246, 2.22305132916493358220373645775, 3.01633102609039308216877472695, 3.262270486773370943046495906003, 4.50655047668956289161334843259, 5.37746103297357766060889682586, 5.72414380581768583991131024589, 6.69560625171755107956842232511, 7.32452366371053828037853584011, 7.93238951637938903481533522424, 8.63028449304847350641038522425, 9.772669672211944183851566797014, 10.08241022944487504717838608982, 10.590339627927592381066591074176, 11.56071312535985799441794412892, 11.976310360074744350968756399680, 12.95409464425496940936041628226, 13.531603801035677623035269790047, 14.13723263736745997742738398134, 14.76701261551039786245668488568, 15.41375872931687913708791763611, 16.028937023531904695566616811116, 16.7844836785530943520266359087, 17.56797416201393099103056891760, 18.08424671281621720864271513339

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