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Properties

Label 1-6048-6048.5683-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  + (−0.819 − 0.573i)5-s + (−0.573 − 0.819i)11-s + (0.422 + 0.906i)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 + (0.906 + 0.422i)29-s + (0.766 − 0.642i)31-s + (−0.965 − 0.258i)37-s + (−0.342 + 0.939i)41-s + (−0.573 − 0.819i)43-s + (−0.766 − 0.642i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 − 0.573i)5-s + (−0.573 − 0.819i)11-s + (0.422 + 0.906i)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 + (0.906 + 0.422i)29-s + (0.766 − 0.642i)31-s + (−0.965 − 0.258i)37-s + (−0.342 + 0.939i)41-s + (−0.573 − 0.819i)43-s + (−0.766 − 0.642i)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: $\chi_{6048} (5683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.468 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4851360216 + 0.2919795140i\)
\(L(\frac12)\) \(\approx\) \(0.4851360216 + 0.2919795140i\)
\(L(1)\) \(\approx\) \(0.7431196856 - 0.09425434118i\)
\(L(1)\) \(\approx\) \(0.7431196856 - 0.09425434118i\)

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 \( 1 + (0.422 + 0.906i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.906 + 0.422i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.573 - 0.819i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.819 + 0.573i)T \)
61 \( 1 + (0.0871 + 0.996i)T \)
67 \( 1 + (-0.422 - 0.906i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.906 - 0.422i)T \)
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

−17.69108001120987717710134104313, −17.13007582334263375297706761553, −15.92375489356980950083009069505, −15.59466879367949962373752315401, −15.18829416788606595516379379521, −14.412288178897202707939753264806, −13.68494588003465956140348001607, −12.75863129555337715054545932913, −12.521701044318581234525580080578, −11.53949308796241427833347471447, −11.01325076190744525352432036655, −10.20316650196924812067474176409, −9.98225650492403776181934076463, −8.59074046741179910142499496615, −8.256743124444040028757334778, −7.61954987726781048385778906983, −6.72112489769650630159904993377, −6.32908109181372229548216529336, −5.266247448530913360872393459, −4.56548832842584340748789189343, −3.83591273035864997484992930840, −3.14623098505224622979067913693, −2.36096374940821331992084208860, −1.51669539283256990257722082726, −0.2019138658338233699281717865, 0.67915801077759099817836070557, 1.70668675647039458810472950990, 2.616197220361548056251421582571, 3.4296052804437704573725971519, 4.25926298407376092617404494635, 4.70206202141760983725177685323, 5.560084702812548036145935316278, 6.459569519070926895649829730033, 6.97180408432736216974355075397, 7.98358962176240072405317855828, 8.50589365291193963636417006047, 8.874693291241465794449815071945, 9.882185249842042566214857611573, 10.62896269597313919492634815477, 11.32013296898957728241406463577, 11.84834224323026976450601796415, 12.47876580582574577304992663846, 13.28603417456599013602363143089, 13.80447604031490048731373125992, 14.5057784908978548737140523263, 15.46971369776050112426600970624, 15.80242264768891102970546999235, 16.60461999155886164186949787161, 16.803752668494742796841861402355, 17.94373448169128555099829060742

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