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Properties

Label 1-6048-6048.1627-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.419 - 0.907i$
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  + (−0.819 + 0.573i)5-s + (−0.573 + 0.819i)11-s + (−0.996 + 0.0871i)13-s + 17-s + (0.707 − 0.707i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (−0.0871 + 0.996i)29-s + (−0.939 + 0.342i)31-s + (0.258 − 0.965i)37-s + (−0.642 + 0.766i)41-s + (−0.422 − 0.906i)43-s + (0.939 + 0.342i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)5-s + (−0.573 + 0.819i)11-s + (−0.996 + 0.0871i)13-s + 17-s + (0.707 − 0.707i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (−0.0871 + 0.996i)29-s + (−0.939 + 0.342i)31-s + (0.258 − 0.965i)37-s + (−0.642 + 0.766i)41-s + (−0.422 − 0.906i)43-s + (0.939 + 0.342i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.419 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04678813410 - 0.07319021250i\)
\(L(\frac12)\) \(\approx\) \(0.04678813410 - 0.07319021250i\)
\(L(1)\) \(\approx\) \(0.7219228250 + 0.1458501770i\)
\(L(1)\) \(\approx\) \(0.7219228250 + 0.1458501770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.819 + 0.573i)T \)
11 \( 1 + (-0.573 + 0.819i)T \)
13 \( 1 + (-0.996 + 0.0871i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.0871 + 0.996i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (-0.422 - 0.906i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.0871 + 0.996i)T \)
61 \( 1 + (-0.906 + 0.422i)T \)
67 \( 1 + (-0.573 - 0.819i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.0871 - 0.996i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−17.99023231842642652266598759453, −16.9523003281208577395429053118, −16.63493502190032373043477123445, −16.04585453735630457765642241877, −15.350727955692380087688823453362, −14.66727607530484539617693179372, −14.03896450325026905322013387689, −13.244309051948559750944365627028, −12.55882670823522334656693988813, −11.96387760414928750666972995558, −11.52609415128851417578271400637, −10.59443562079389933311407976984, −9.94506608221737731172992514845, −9.30668564938854567879634291402, −8.31973369039307272733573793152, −7.9351166725941542449670556510, −7.42908729693251230469031383091, −6.415275126549630558054590588051, −5.52151197303656772028366230556, −5.10429694473439128678193936240, −4.19762248739722329198150781767, −3.50021098283216227006543257, −2.82711617007585582727164711988, −1.82025167852703211666103844033, −0.80300456819338010498532370820, 0.02803882081747451505724983179, 1.32405929817504393584820874057, 2.29782969435713330336176937446, 3.02438210186792438978244713632, 3.66000377235006371485742191745, 4.56018055199168501799609564134, 5.174883951228938809328642029472, 5.91799502855279259843447124155, 7.10265347724722181309134508915, 7.363727498116968088412501860337, 7.81342633424548412538572249592, 8.87970047824377191836617177647, 9.57090195158746655314640074025, 10.28078955221420421922648850106, 10.80273380129954995415209129914, 11.69349834190009544957617878102, 12.18559751059228963192172322938, 12.68747869632619055013472674794, 13.67319315517782376410558145880, 14.346743948396843064359912776417, 14.92968342667145826597671472404, 15.46301176754309916060606624785, 16.1217429011912271975952467193, 16.76603807520633362408358001350, 17.604163932651994299289763808944

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