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Properties

Label 1-6048-6048.3197-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: $\chi_{6048} (3197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.999 - 0.0322i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.959512077 - 0.03156237044i\)
\(L(\frac12)\) \(\approx\) \(1.959512077 - 0.03156237044i\)
\(L(1)\) \(\approx\) \(1.181007410 + 0.07829205204i\)
\(L(1)\) \(\approx\) \(1.181007410 + 0.07829205204i\)

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

−17.76748484780651562071471672948, −17.04702231665206967454189925260, −16.334301495614816131438820144454, −15.902493245277883885072999142176, −15.18066220845507966375027080995, −14.28256868110560542279331213623, −13.76999450468766408306117324210, −12.89603990058428364264719878111, −12.77423254675788540481555361453, −11.78982064304096710313313444019, −11.08572257803428084141885844150, −10.39527830546514500222542589889, −9.61195876208915177740572421296, −9.08547613361784078209572810972, −8.49476402216491497792541667548, −7.61278586663687747251039828412, −7.02223226797790846617012021421, −6.02124624888121294231918223761, −5.560437490758741502853210056242, −4.68024852734874106046529460133, −4.27175914696603752272072884674, −3.12598753255020777128981974657, −2.364072477018769193609650244366, −1.574668955961152555938625848961, −0.788838938878127807475689510057, 0.62589078181424449385421703409, 1.724226840241278838938507686141, 2.442425553330924616832568357087, 3.25723343932847337678723641956, 3.74181797672798370021450183333, 4.900350499479723294044376967150, 5.59603773506393873964436640247, 6.26445564994587300878768179806, 6.71972368136023105979944179987, 7.72918765600229978501394492654, 8.38206229189432632737024972154, 8.863856763119417893734324005414, 10.092577105260813763381637668616, 10.39597097135288623006036199004, 10.868139245287655004616128706, 11.64843336056264329239505797580, 12.67915233669458453314914821104, 13.07686851757347434888065169508, 13.78233512958558302978267011889, 14.44878096682497268639082387426, 15.043378324905700700839060187510, 15.679317985687780852233524582942, 16.37453891407948592950841662622, 17.187302054537631404759877898011, 17.69556421996854573576165230149

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