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Properties

Label 1-6048-6048.347-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  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (0.0871 − 0.996i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (0.996 − 0.0871i)29-s + (−0.766 + 0.642i)31-s + (0.707 + 0.707i)37-s + (0.642 + 0.766i)41-s + (−0.906 − 0.422i)43-s + (−0.766 − 0.642i)47-s + (0.258 − 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)5-s + (−0.0871 − 0.996i)11-s + (0.0871 − 0.996i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (0.996 − 0.0871i)29-s + (−0.766 + 0.642i)31-s + (0.707 + 0.707i)37-s + (0.642 + 0.766i)41-s + (−0.906 − 0.422i)43-s + (−0.766 − 0.642i)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: $\chi_{6048} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.157 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6113069144 + 0.5216645969i\)
\(L(\frac12)\) \(\approx\) \(0.6113069144 + 0.5216645969i\)
\(L(1)\) \(\approx\) \(0.8175053061 + 0.004086609175i\)
\(L(1)\) \(\approx\) \(0.8175053061 + 0.004086609175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.996 - 0.0871i)T \)
11 \( 1 + (-0.0871 - 0.996i)T \)
13 \( 1 + (0.0871 - 0.996i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.996 - 0.0871i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (-0.906 - 0.422i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (-0.573 + 0.819i)T \)
61 \( 1 + (-0.996 + 0.0871i)T \)
67 \( 1 + (-0.906 + 0.422i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.996 - 0.0871i)T \)
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

−17.664017445901428387207929345, −16.742763473900870848536553289773, −16.12600960003883536268707670539, −15.72478865775620240976730162481, −14.96489102035745782769172691041, −14.31254116915522963100023042405, −13.764735305719178331111106153157, −12.75853508862560045532922842098, −12.28396496347428022583726387874, −11.567133587554715539230189830236, −11.14888388031692063213730330360, −10.29218988435554862801601268711, −9.43071208833009389506483211826, −9.01804512544515090825519948301, −8.05530657185019576513400686764, −7.44020539499509734098369962342, −6.919286341541011297456499349422, −6.23031578312928114665803090760, −5.07740177601947591589083548143, −4.49640791317126635743538093735, −4.02921426666205181119686702968, −2.991211467537573213728919283, −2.36089341512441998534314076639, −1.36418912270681471372115065400, −0.26196482325655695333063712372, 0.83208537682915759889050327118, 1.60718483536888711158471322138, 2.99003136241462572104514584585, 3.29212714593120690351513868225, 4.06640640878757103531109450571, 4.94365966761985640848890209324, 5.62888123458898183594955223209, 6.34988466969457426710638430235, 7.20151849840273413085990352459, 8.04770656354058093556208643411, 8.23480628840631193447611218839, 9.07475167391426689291706156201, 9.98586598122362692223572757267, 10.68051515280189200702959343637, 11.26580497225411627548133204073, 11.892448911438587367616190223500, 12.5141177884453789541433649068, 13.3445934372385552061146454623, 13.75314360398482512402117916411, 14.87463319276938407680010570978, 15.132648672171325138138715986493, 16.070602947319808903007369140683, 16.26924570788330767862697541495, 17.14583488664741664692805801938, 18.04548148058642986768508989585

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