↑

Properties

Label 1-6048-6048.5731-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.527 + 0.849i$
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  + (0.996 − 0.0871i)5-s + (−0.0871 + 0.996i)11-s + (0.0871 + 0.996i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (0.996 + 0.0871i)29-s + (0.766 + 0.642i)31-s + (−0.707 + 0.707i)37-s + (0.642 − 0.766i)41-s + (0.906 − 0.422i)43-s + (−0.766 + 0.642i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.996 − 0.0871i)5-s + (−0.0871 + 0.996i)11-s + (0.0871 + 0.996i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (0.996 + 0.0871i)29-s + (0.766 + 0.642i)31-s + (−0.707 + 0.707i)37-s + (0.642 − 0.766i)41-s + (0.906 − 0.422i)43-s + (−0.766 + 0.642i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.527 + 0.849i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.527 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.091102197 + 1.162802967i\)
\(L(\frac12)\) \(\approx\) \(2.091102197 + 1.162802967i\)
\(L(1)\) \(\approx\) \(1.332682698 + 0.2150971757i\)
\(L(1)\) \(\approx\) \(1.332682698 + 0.2150971757i\)

Euler product

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

−17.61782939032346975935979722399, −17.0515217308076306390302963085, −16.28465136821759153867797729062, −15.72182434142598553156942341802, −14.90914717167937508456347886000, −14.184414742701140937687653147030, −13.771703359233809142502910553913, −12.86017713085196703781478218255, −12.69947602783029318073530490151, −11.48385311392507518442139686992, −10.97102071846355218771520112556, −10.162439443153299914151199454152, −9.86682659469886933075139149406, −8.74994377942454666743586299518, −8.43223442954781692481403401355, −7.58606778748479868069471215538, −6.591786815309909994234385909980, −6.08646703532620426870493363238, −5.49496427397700242975865234826, −4.77997040622329633597857866714, −3.7833858186127467022486641632, −2.946458877724613012989024564511, −2.44895045041902261376668739521, −1.37394041580210613714919472735, −0.64811214070291975685303197999, 1.06277405800079736996217162015, 1.69955213814763167632604737651, 2.54192116858008074269680860295, 3.146600548521132034836287277318, 4.40472830937434293105066142873, 4.81245854678832052489505170150, 5.565637247722945852488643240667, 6.389320709957886494484705525303, 7.07229882560788047783137263850, 7.50433839133202775014997320825, 8.80874407794862592418196142049, 9.10944599917738841700416941999, 9.85795633579547037068682438605, 10.32265050443750456369199737371, 11.28276186413692563070255245518, 11.90117645335665270700097553318, 12.53767469670836219275424197331, 13.36642247132318195897958443436, 13.99092837582312954573213624146, 14.19601865118711122413337202090, 15.356844347010654409593500966098, 15.78589824055064757118758169201, 16.52598311758530464913950294081, 17.359455144958414998223842548220, 17.73657855231841417103484766805

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