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Properties

Label 1-6048-6048.2117-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.995 + 0.0902i$
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  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.573 + 0.819i)13-s − 17-s + (−0.707 + 0.707i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.819 + 0.573i)29-s + (−0.766 − 0.642i)31-s + (0.258 − 0.965i)37-s + (−0.984 − 0.173i)41-s + (−0.996 − 0.0871i)43-s + (−0.766 + 0.642i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.573 + 0.819i)13-s − 17-s + (−0.707 + 0.707i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.819 + 0.573i)29-s + (−0.766 − 0.642i)31-s + (0.258 − 0.965i)37-s + (−0.984 − 0.173i)41-s + (−0.996 − 0.0871i)43-s + (−0.766 + 0.642i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.995 + 0.0902i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.995 + 0.0902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028901348 + 0.04653256166i\)
\(L(\frac12)\) \(\approx\) \(1.028901348 + 0.04653256166i\)
\(L(1)\) \(\approx\) \(0.8206782066 - 0.04874315405i\)
\(L(1)\) \(\approx\) \(0.8206782066 - 0.04874315405i\)

Euler product

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

−17.86307776551263745903819760658, −17.08740465481899211349423828255, −16.26722719320733889418252025582, −15.540208279879967697516545293, −15.10616991389734409843578316782, −14.80040863674617464402419588166, −13.5436575898731270967656627144, −13.153679401080419877301110668151, −12.44693127857859297572426533370, −11.69087121289890487988986832768, −11.075431331540756606336570387086, −10.47773120715234512467235871948, −9.91839800565594617961804841692, −8.711180681127139112670995191236, −8.47956582925222577845351341347, −7.587684798290591274105536565288, −6.858862312068416072333633176058, −6.5168998208173167940013714106, −5.31763745447570725732779810870, −4.664460036686079279233459489, −4.076248087840652041366799345935, −3.09527242999753990627565837114, −2.63130854328892982635134171775, −1.59107806559119551649985829267, −0.451504630929032551864925857927, 0.5624301503741425415930295992, 1.557386806257954107599910893068, 2.420272391576297334393547753470, 3.564519263857420639946778351800, 3.795755344218631824688311682635, 4.82344599564558066009202875744, 5.31068821261751491907628126603, 6.454537315596673839030082340033, 6.78246029034858899983763031817, 7.85506424847176090131628035067, 8.37731364067995120963595489094, 8.889948233878444419612902068124, 9.58928622603274708103898924428, 10.806059665513768729245336029111, 11.01647348006933191593193967932, 11.69358827147834648135492623016, 12.4893852488162695057447176212, 13.1241101467823785713099007983, 13.646285451319983189790805235142, 14.56485530477439576399861157377, 15.115264142565054035847483331021, 15.9218985209608968467177228751, 16.32569799311814203717547254975, 16.849518877382739465982030431663, 17.71285481799572218330336345546

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