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Properties

Label 1-6048-6048.3715-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.999 + 0.0322i$
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  + (−0.573 − 0.819i)5-s + (−0.819 − 0.573i)11-s + (0.819 − 0.573i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.573 + 0.819i)29-s + (0.173 − 0.984i)31-s + (−0.707 + 0.707i)37-s + (−0.984 − 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.573 − 0.819i)5-s + (−0.819 − 0.573i)11-s + (0.819 − 0.573i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.573 + 0.819i)29-s + (0.173 − 0.984i)31-s + (−0.707 + 0.707i)37-s + (−0.984 − 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 + 0.0322i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.999 + 0.0322i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01404846435 - 0.8721821327i\)
\(L(\frac12)\) \(\approx\) \(0.01404846435 - 0.8721821327i\)
\(L(1)\) \(\approx\) \(0.8042711376 - 0.3274056114i\)
\(L(1)\) \(\approx\) \(0.8042711376 - 0.3274056114i\)

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 \( 1 + (0.819 - 0.573i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.573 + 0.819i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (-0.0871 + 0.996i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (0.422 - 0.906i)T \)
61 \( 1 + (0.573 - 0.819i)T \)
67 \( 1 + (-0.0871 - 0.996i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.573 - 0.819i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
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

−17.93872757754801432656769963325, −17.65181879777584254576362145482, −16.63337877525316196355743939930, −15.85952462529152913310115163713, −15.49029932843683480520236582281, −14.87582866366491817574050333150, −14.11496889452404208892066158535, −13.46572200241004543743973735638, −12.85268803384472237413698451610, −11.93266583702278332726342461528, −11.46884312511198761346097663603, −10.72799454350591617192331005690, −10.24365442312384256456991797524, −9.452393616807710678013359200420, −8.5566318274184991499300718973, −8.01626774895206851150900290527, −7.09288739202975027066572912114, −6.88278595819304184305830961809, −5.80901009479592273180092591024, −5.22669407451060243115659790029, −4.15200337458162281366386189262, −3.69606895669038443099382465340, −2.88712100997744615030486470579, −2.0735105469410389499088368105, −1.22748469802573693485962825373, 0.261186833920449079490703072591, 0.94086106324171097477827569698, 1.944210646192295310966635479851, 3.12194477621293643886333772758, 3.390372815312425516008103350947, 4.5684639224265345525701852914, 5.06337636065832191254656242189, 5.670479062429666916277218816411, 6.613306263135924708942367971307, 7.42661933265977198327082900644, 8.055605129745913156268184177690, 8.66451501467529468273117344067, 9.21785154750955963086161473403, 10.07786419022664866943181088623, 10.937454210303596714803210246511, 11.36721535331709665268412525419, 12.099092628525014729299412960980, 12.869339630619628435949030810921, 13.42798639298468670155403134640, 13.81938513477683301609956837808, 15.046214735735768784272640916801, 15.42442563606743949864217677859, 16.211555110356575572441503176034, 16.44904285319498933165624509727, 17.323735625712187586801347287342

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