↑

Properties

Label 1-6048-6048.115-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.735 + 0.677i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.573 + 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.0871 − 0.996i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.996 − 0.0871i)29-s + (−0.939 + 0.342i)31-s + (−0.965 − 0.258i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (0.939 + 0.342i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.573 + 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.0871 − 0.996i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.996 − 0.0871i)29-s + (−0.939 + 0.342i)31-s + (−0.965 − 0.258i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (0.939 + 0.342i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.223000333 + 0.8684760262i\)
\(L(\frac12)\) \(\approx\) \(2.223000333 + 0.8684760262i\)
\(L(1)\) \(\approx\) \(1.309159614 + 0.2250469494i\)
\(L(1)\) \(\approx\) \(1.309159614 + 0.2250469494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.573 + 0.819i)T \)
11 \( 1 + (0.819 + 0.573i)T \)
13 \( 1 + (-0.0871 - 0.996i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.996 - 0.0871i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (-0.906 + 0.422i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.996 + 0.0871i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.939 + 0.342i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.494106253808322668347120276377, −16.74151140633301053566011079048, −16.623018676085819007794641450895, −15.82689750397975883872888258606, −14.8758243318210759042196437311, −14.32417324403259133399323349837, −13.55483756133237214860255886921, −13.2499227380609768197080449429, −12.27395773778251280693377898557, −11.71293545505264005061324451888, −11.21123806679288950210383906108, −10.17315703720225307341434424718, −9.48363267573578900299755232250, −9.04277482193576552547657915424, −8.5077136310618672645274740947, −7.41607281909140031480749099358, −6.954062820779472937959802746633, −5.92915843561309313088324377656, −5.477008236268220837116634403634, −4.76128061304016071383458097351, −3.865911872347799781553636262191, −3.27566569342902119591801500739, −2.13523794485515286177293737821, −1.45555712916386082904747002062, −0.740405398368982830488479680526, 0.88923151462604235331274600354, 1.757339393760673584455944097493, 2.5010207021623917519861979390, 3.45311085423845422673958134514, 3.76503972557805082064987643758, 5.10443270865241841093099962024, 5.54394339285260939246973640707, 6.28968602033922847487402485859, 7.1315314346909743152550953082, 7.50423080757516540853115741268, 8.413589751326824064699710339280, 9.37498331208357336658371322211, 9.75042087849451580081349695006, 10.57702201544367484584600088789, 10.94285420559417496449903019217, 12.00365309588748335204044543897, 12.43584403594154273208738438879, 13.20121628958769057045500934233, 14.02906909393599473593018489159, 14.55541755427174536557843309031, 14.95341583973233990388966716409, 15.73461605692058323989178885738, 16.68370853092444871984709094726, 17.09224460786538006169662385320, 17.85085816898224414058446000516

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