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Properties

Label 1-3024-3024.1453-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.925 + 0.377i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
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

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

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.701897773 + 0.3339330276i\)
\(L(\frac12)\) \(\approx\) \(1.701897773 + 0.3339330276i\)
\(L(1)\) \(\approx\) \(1.157061259 + 0.03427818781i\)
\(L(1)\) \(\approx\) \(1.157061259 + 0.03427818781i\)

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

−18.79745720115069701070338905988, −18.2426209356438907746319885499, −17.699198238360744922325572049613, −17.03659554100252156813716046991, −16.16225580263478926275408148812, −15.622941241361971069325439160108, −14.6276260070539873252160739226, −14.13371359013377858727895520841, −13.39536072603734614740567450923, −12.777958251433293514763551080005, −11.98820007777887382202066427047, −11.18433048907685317728477297530, −10.236474783847561330792324041544, −9.84816861252860824505555357202, −9.235574407870360005169748251909, −8.050777451158093785180412864089, −7.607742661527870436463389411519, −6.551328195987710222635983036103, −6.013006704485971329205976977199, −4.92834234325694517780788207443, −4.727052110437980467581890966798, −3.18737523249878457409353436626, −2.57996167440966263503571910565, −1.90579713225233811606090535654, −0.63868038767321616197991487254, 0.84284715830884804172654117919, 2.02311307536677151815882164760, 2.494717246459735286875521308990, 3.50824845216145071628276121488, 4.60779153379209417394221158025, 5.31632812329477908421233156910, 5.83969046266652787270692763906, 6.85329656677499770652331301897, 7.47192721066561732106113371189, 8.47648472473507125311060862054, 9.069686585519284345322588479438, 10.00418325734267337807248923243, 10.352380972822451130109961389215, 11.21189056130274685806775559846, 12.2754043861287875684846271455, 12.671988751218464178938447227017, 13.70966059801116218020328333714, 13.91083064081957907996081922540, 14.865911090436301667703170100046, 15.72983606002913440400120704708, 16.209095664078855286111145583930, 17.254201449856292081604850661276, 17.63806182956368927312487232281, 18.192757706867774425949384439, 19.14818612785802115172332745466

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