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Properties

Label 1-6048-6048.1651-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.468 + 0.883i$
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  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (−0.996 − 0.0871i)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 + (−0.0871 − 0.996i)29-s + (0.173 + 0.984i)31-s + (−0.965 − 0.258i)37-s + (−0.642 − 0.766i)41-s + (−0.422 + 0.906i)43-s + (−0.173 + 0.984i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (−0.996 − 0.0871i)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 + (−0.0871 − 0.996i)29-s + (0.173 + 0.984i)31-s + (−0.965 − 0.258i)37-s + (−0.642 − 0.766i)41-s + (−0.422 + 0.906i)43-s + (−0.173 + 0.984i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.468 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175670478 + 0.7075782457i\)
\(L(\frac12)\) \(\approx\) \(1.175670478 + 0.7075782457i\)
\(L(1)\) \(\approx\) \(1.050913940 + 0.02509789466i\)
\(L(1)\) \(\approx\) \(1.050913940 + 0.02509789466i\)

Euler product

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

−17.47350578030637286014696960905, −16.9650257704285649791314192307, −16.5688624000343245412489061196, −15.32984491816655704379733502826, −15.08574460373781326171034472472, −14.28706264862893334019885520873, −13.64737443204374370030865794375, −13.01476731880838406332349606924, −12.56334734430939388841616067391, −11.45608994559726763751584729535, −10.93431050769361929567866005911, −10.30171656869576210477562747858, −9.7355383188725071707495769638, −8.82449868457685364413499492664, −8.43221922200340610884962353523, −7.40424839824269505678956576999, −6.67230098400653681482512096573, −6.22957784085188488585981764175, −5.25551500197814967938939190788, −4.90374494858990553490243567908, −3.70658986642993038119596026194, −3.05069702199454731022401051078, −2.21709810112693564598535330206, −1.68235252886094813065068371775, −0.37173464096374739973058640533, 0.855982200084207262907771909773, 1.978125336618960273799787807911, 2.33795003250125143825179307288, 3.20618762463757996287432238132, 4.45991278973646501664249116335, 4.87553351241720936582751479047, 5.433141076892664649498676120832, 6.48454513546384429556697805030, 6.9159525786211718859957688684, 7.76568213321669127847767335004, 8.544008940955564037608623284104, 9.32697311954324549780704051592, 9.75287565361934774663650443452, 10.44538670795682715020322649786, 11.09370096935394880164804661916, 12.13683572592941322984348113699, 12.60463919016824128647102576450, 13.10074127228418072971784985769, 13.9757516810221238297333429578, 14.39846929212599147928824251460, 15.32859251226739044497257360959, 15.69703375676893603332945038081, 16.782409581043941119837966054861, 17.159629086473078379461609677673, 17.68358811591651065517942667195

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