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Properties

Label 1-6048-6048.4469-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.527 + 0.849i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.527 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6768565291 + 1.217210666i\)
\(L(\frac12)\) \(\approx\) \(0.6768565291 + 1.217210666i\)
\(L(1)\) \(\approx\) \(1.025146038 + 0.2886944987i\)
\(L(1)\) \(\approx\) \(1.025146038 + 0.2886944987i\)

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

−17.43517990856591017559665854530, −16.81366714032718406237524237087, −16.41147809011736680578838497279, −15.51434719794108006449829705057, −14.828411442547042646914683270838, −14.13745652626577784682450547939, −13.64756079789070475045235891238, −12.658225797783806530981390885250, −12.33920396434781847133238732727, −11.79787924335824957430398066530, −10.67865012152104237698471825840, −10.18070225478068562339061548069, −9.50059941760393275944828567719, −8.710624068978653700952436937095, −8.29188692227868513826592364406, −7.51908867110106825475088178375, −6.46067398847589204128106262996, −6.02813687651466172444485302737, −5.25524681740594912317404010450, −4.47374328515953117665052750858, −3.89822353346557102492369897022, −2.94577299354830485089060006957, −1.97198368446131406374923581313, −1.37560868001649811504490263450, −0.35828379993304789827992139035, 1.0456210273096625456616977929, 2.18504988729392489431485222288, 2.45124808389039901113665751825, 3.50273152130522524004456951081, 4.24216941979061993889960519868, 4.94052727970850794411956176960, 5.93261339051521856401933488756, 6.4761638762024188180577349720, 7.14151808475933517419170591304, 7.675608024835738099431409880462, 8.67823733872492087612497543458, 9.46049312894515277045227420102, 9.93630054623876022780999670715, 10.5216652329085458067457440923, 11.42992373854701313616100091231, 11.955095359248980958370810859407, 12.51422065454096125305880661819, 13.5577815430879671671356461520, 14.19960053609715777913092607509, 14.40829929547513188582421200599, 15.3139268009117219815640695158, 15.80876318057902710308333834700, 16.832805982515122406435995947, 17.37375975088359784011839658620, 17.711930660153054989432061746519

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