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Properties

Label 1-6048-6048.947-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.157 - 0.987i$
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  + (0.422 − 0.906i)5-s + (0.906 − 0.422i)11-s + (−0.906 − 0.422i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.984 + 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (0.939 − 0.342i)31-s + (0.707 − 0.707i)37-s + (0.342 + 0.939i)41-s + (0.819 + 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.422 − 0.906i)5-s + (0.906 − 0.422i)11-s + (−0.906 − 0.422i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.984 + 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (0.939 − 0.342i)31-s + (0.707 − 0.707i)37-s + (0.342 + 0.939i)41-s + (0.819 + 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.157 - 0.987i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.733443461 - 1.479250542i\)
\(L(\frac12)\) \(\approx\) \(1.733443461 - 1.479250542i\)
\(L(1)\) \(\approx\) \(1.218342214 - 0.3865469547i\)
\(L(1)\) \(\approx\) \(1.218342214 - 0.3865469547i\)

Euler product

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

−17.7537564798345501040213634822, −17.22840627804495679200835232841, −16.7731728763864683289879358175, −15.77457184199993212863803731927, −15.07968240547268132245728466525, −14.55696206152336398515974205724, −14.093416956295460732056981212921, −13.3485261164284686388273251203, −12.515884354872057871569275905727, −11.92529394336437321925461334000, −11.20096574714953117624072998752, −10.575880907976099572914928378961, −9.83971265286503589143757222052, −9.34387393941291584644860145892, −8.60992023774569017291294685413, −7.61505759513232403455400937147, −6.90630927598514133703015869215, −6.640494053593137482663545029393, −5.67207008611523283540550778379, −4.979611417989061765026153345981, −4.063477004208725299552017210081, −3.423013158218350715230495734275, −2.517402284052133721312585409449, −1.92357445174237643627024630187, −0.99560462558526094659423135315, 0.728331541397339218196190554335, 1.11016526377770867367894338652, 2.39482211240028146323584330525, 2.83112217245041339929394262479, 4.0478835643857401629120852722, 4.56523990764296794610875573416, 5.38375429246060996117939092, 5.87822140254588509754400043987, 6.81309777486647405073935270157, 7.49637735261022066074934108727, 8.2256639116913438008939494617, 9.09776027994905530846287692203, 9.47877713477524884292399522385, 9.988176606022749257923422577182, 11.21499686380580379644077788443, 11.53802825182039265660258084967, 12.36292852386188723784448609941, 12.94728691629334231071866189939, 13.66533620972861379741268470588, 14.12806945593839640282531494225, 14.958547137055063064287576738952, 15.69151048063680114485066247213, 16.29527449690330683541453737531, 17.033375473880712860773022042040, 17.37984782246911884784740330804

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