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Properties

Label 1-6048-6048.2189-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.849 - 0.527i$
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 + (0.906 + 0.422i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (0.939 − 0.342i)31-s + (0.707 − 0.707i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.422 + 0.906i)5-s + (−0.906 + 0.422i)11-s + (0.906 + 0.422i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (0.939 − 0.342i)31-s + (0.707 − 0.707i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.211251876 - 0.3455553026i\)
\(L(\frac12)\) \(\approx\) \(1.211251876 - 0.3455553026i\)
\(L(1)\) \(\approx\) \(0.9474898429 + 0.1057342765i\)
\(L(1)\) \(\approx\) \(0.9474898429 + 0.1057342765i\)

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

−18.01309619164078705343524698871, −16.928771970968485294126101367778, −16.18225944122266628498002115838, −16.07490715489808151684128442226, −15.33661046003165663666147145172, −14.47443382374579863006158300693, −13.51879077751216523194693303601, −13.365867928236772911350817694846, −12.50112081730863236001047798177, −11.72419202997261365068678221323, −11.417124754549076297428957339663, −10.3508927274634222146277139441, −9.8570826466222826624434388116, −9.01830896859154179153362217029, −8.24212485611681004793314086188, −7.9473341263338914697318785398, −7.14410086975231589444318952637, −6.122376496520154030638750688133, −5.39489723585231034832856108444, −5.00748018556625653983604351226, −4.031524069745046267128345095636, −3.31589301052848935287081919631, −2.65678068738868635523413610446, −1.37304526014008390383495788467, −0.8799258249231594764498603002, 0.38865793324331014520905596914, 1.65575310934176510951573992970, 2.40124982645412116189757939398, 3.18092870460229431070773554592, 3.91762761031519573366646457277, 4.489688779604600115966635909, 5.68735410400250822861695496134, 6.04341453697033581668901664908, 6.94918179952568782844377012877, 7.67478074599770828897537528922, 8.04816683232344869578873677935, 8.96017344441047073733841937857, 9.865153302156019057419526787935, 10.41063912756258351517171741495, 10.930853797173884660062305380446, 11.82660390236549891688830238424, 12.12764010383725716858092966226, 13.25605790554525296378329452733, 13.66231548206188260213120404877, 14.41303681723917273811611863934, 15.108397034131736657374822192990, 15.692184217035220716396369842398, 16.10692604538046102168133755553, 17.04674512893912717619477223104, 17.80365075402612043526326733386

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