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Properties

Label 1-3024-3024.221-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.377 - 0.925i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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  + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)11-s + (−0.984 + 0.173i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.984 − 0.173i)29-s + (0.173 + 0.984i)31-s − i·37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.173 + 0.984i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)11-s + (−0.984 + 0.173i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.984 − 0.173i)29-s + (0.173 + 0.984i)31-s − i·37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.173 + 0.984i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.377 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359253380 - 0.9133424569i\)
\(L(\frac12)\) \(\approx\) \(1.359253380 - 0.9133424569i\)
\(L(1)\) \(\approx\) \(0.9528864086 - 0.06506816698i\)
\(L(1)\) \(\approx\) \(0.9528864086 - 0.06506816698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (0.984 + 0.173i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.984 - 0.173i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−19.12716087821891952430664048595, −18.517776270965399678747239483385, −17.393988585616710937128353061994, −16.890978941388007452830376127991, −16.35215908734084766128165434344, −15.35306851595758417333926270487, −14.89684322270781395910374029388, −14.29371734902186297817067991377, −13.284124560163861365483745192180, −12.56659369055685397786612173528, −11.81500448582228425521730747990, −11.4936736509304672884304379769, −10.49617695852883929297787084132, −9.67792519514788587647114767903, −9.0023488507669195771046097819, −8.149589728634510104198842052930, −7.482970709328648780788323224071, −6.91965153741387071723355956564, −5.81132298814673708313852608503, −5.15502288180007945749944195166, −4.106617225542103026118940076106, −3.6571964748723075752930458758, −2.75725195933366721639648503954, −1.53097975988908511315469889853, −0.75937404849354020928658130480, 0.37520020946673040613235917790, 1.161803962901332815037792017132, 2.44695712885853015298761820187, 3.17981431297028484114601784514, 4.01303777614558783783231842904, 4.76498633429399343443887764117, 5.46493590186340671064433056239, 6.73681488040818238632863735755, 7.16281252508839475922818997488, 7.77396874801423563607275518861, 8.821502198683541537434497977513, 9.374654322200623559022384242478, 10.16266241782048305873366929898, 11.2081154258097153593669953417, 11.611226638340595446612501039167, 12.306980704638806210428327736164, 12.93212490236277174456622525657, 14.13936604672664428732477369584, 14.49548096314828779593574789321, 15.182432979708971471972977053491, 16.06809783343293931331325169138, 16.50605943248673299170131923762, 17.354061150150638773099992139309, 18.04617597860736015879564478654, 18.96439881511623280972433303455

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