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Properties

Label 1-6048-6048.4811-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.526 - 0.850i$
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 + (−0.819 − 0.573i)13-s + 17-s + (0.707 − 0.707i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.573 + 0.819i)29-s + (−0.766 + 0.642i)31-s + (0.965 − 0.258i)37-s + (−0.984 + 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (0.965 − 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)5-s + (−0.906 − 0.422i)11-s + (−0.819 − 0.573i)13-s + 17-s + (0.707 − 0.707i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.573 + 0.819i)29-s + (−0.766 + 0.642i)31-s + (0.965 − 0.258i)37-s + (−0.984 + 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (0.965 − 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.526 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6277321923 - 1.127477214i\)
\(L(\frac12)\) \(\approx\) \(0.6277321923 - 1.127477214i\)
\(L(1)\) \(\approx\) \(0.8908004799 - 0.3028415888i\)
\(L(1)\) \(\approx\) \(0.8908004799 - 0.3028415888i\)

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 \( 1 + (-0.819 - 0.573i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.573 + 0.819i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (-0.0871 - 0.996i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
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 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.573 + 0.819i)T \)
89 \( 1 - iT \)
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

−18.07581324676470381810550068036, −17.277676573572021943121889196162, −16.55634746996645780845068525154, −15.971973290508712844987350029687, −15.16555855673611199884479961073, −14.68961843831049834632595637644, −14.20532519697849559957598367946, −13.29204223197389456155278033332, −12.69539323789118046303747859621, −11.70126469114338431735021562148, −11.5950463327564280496311492061, −10.53808917577681733611434212672, −9.96309824107023334952705546494, −9.56360114993728663152001519692, −8.40716386033663320014876871641, −7.63445513729277037127855522776, −7.39353245650582993552583664669, −6.55341668204041106170652078966, −5.722666804770197478723800194370, −5.01557049876249639265863812681, −4.22788555307177233696670511455, −3.38460432560126429795624084916, −2.74148017142081662861086257538, −2.074724154205650156973901828441, −0.93519584945960148022196646004, 0.42151506131306618133871778768, 1.06599518112099713194177178180, 2.18063767483349563413112829896, 3.10969648283232639103046378698, 3.57388223973442675315035437647, 4.815783264669093366751550811193, 5.16461317377495112174642041862, 5.629328002355582595043935078519, 6.95930787495456506879468388104, 7.37894417574162726192267486423, 8.25862147273727640197148205583, 8.61062814929522014589111169995, 9.554410451531731753200941235599, 10.099521209145727013819996730092, 10.92034832392260440381832456393, 11.59692320216206938623523889647, 12.32629855704252766458646983017, 12.85819867355717215067585426673, 13.35882531617937920018171916697, 14.24248454367421635034196419542, 14.95365193072874111783640216838, 15.589392091777243426839079541836, 16.24940316843357307645752035512, 16.7009564746300858457760620909, 17.41239078568263977404571513240

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