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Properties

Label 1-6048-6048.1109-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.907 - 0.419i$
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  + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.422 + 0.906i)13-s − 17-s + (0.707 − 0.707i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (−0.173 + 0.984i)31-s + (−0.258 + 0.965i)37-s + (0.342 + 0.939i)41-s + (−0.573 + 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.0871 − 0.996i)5-s + (−0.996 − 0.0871i)11-s + (−0.422 + 0.906i)13-s − 17-s + (0.707 − 0.707i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (−0.173 + 0.984i)31-s + (−0.258 + 0.965i)37-s + (0.342 + 0.939i)41-s + (−0.573 + 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.907 - 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145647483 - 0.2521077805i\)
\(L(\frac12)\) \(\approx\) \(1.145647483 - 0.2521077805i\)
\(L(1)\) \(\approx\) \(0.8822121759 - 0.1293024852i\)
\(L(1)\) \(\approx\) \(0.8822121759 - 0.1293024852i\)

Euler product

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

−17.790568795337469801569958765847, −17.387842703484427881360985826149, −16.24209925833317906503737406035, −15.60541434577670873623253820290, −15.33068940939039690043623551778, −14.4174016015730085291545908786, −13.928370395091189465077480824290, −13.14225023700714107231120910832, −12.571405684558948772451592162312, −11.68709515236210887344427292701, −11.1287955210812422088198403687, −10.3309891673492920976649935515, −10.06610782557624897954660843580, −9.15976754927464818257493769391, −8.20564621029293898975642218081, −7.591526760785917950835810490139, −7.17268955559140995855828980386, −6.218642237223828364004373152000, −5.57733921183090631489382513166, −4.9269442606885296516727592302, −3.86282668437965673506829987108, −3.27347664330243343643385306998, −2.477376414008114251807368281870, −1.91982957145617539181470588662, −0.511463319727999599881395659255, 0.53374151783441953229558897782, 1.5215300599639724347596198126, 2.36791244319144600454661533221, 3.05963908247816477610788131399, 4.210299596428085394866322458640, 4.77920237600658016187282191358, 5.15179038799035145179216458367, 6.24154015046425875286657529687, 6.8357885190760803611838400431, 7.69946191916199538166319583222, 8.424860137333130592911301805, 8.86504784305483812819899032971, 9.68781704172051832473119099827, 10.24925692818424659593216721802, 11.19622783786115614703561856093, 11.736666674982503743357103289667, 12.44726044766660004606786862759, 13.06183736581713739119150856550, 13.66556092746819807946923048116, 14.22753400968779759571201827687, 15.28462447360978245192527992905, 15.7223564153065616506050343327, 16.38291252032555949221868443647, 16.84966239364995064354914936678, 17.751772048740385695956621759414

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