Properties

Label 1-3024-3024.2525-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.947 - 0.318i$
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  + (−0.642 − 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s + 17-s i·19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (−0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (0.939 + 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.173 − 0.984i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s + 17-s i·19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (−0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (0.939 + 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.173 − 0.984i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.947 - 0.318i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04753453653 - 0.2907363365i\)
\(L(\frac12)\) \(\approx\) \(0.04753453653 - 0.2907363365i\)
\(L(1)\) \(\approx\) \(0.7678822623 - 0.08037093440i\)
\(L(1)\) \(\approx\) \(0.7678822623 - 0.08037093440i\)

Euler product

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

−19.35537092857468356036763611180, −18.59639820698875992481051732465, −18.17854720018336531556819928649, −17.32891384683493197945010987446, −16.42634713885485521274092382200, −15.81835777862761326304640582624, −15.27985899816941460214047287006, −14.22933770970003264309824297357, −14.13632926468209473183812019389, −12.88260730323162044148542574317, −12.322460190042887288758756104515, −11.56283431525139983369115289159, −10.79051214567309530276109198622, −10.2349311566598809274097592521, −9.578567870009655082732283416360, −8.23116528448129315812992289834, −7.93919901254243213259591932058, −7.32977919516318455928080497316, −6.11056895236765057242113577866, −5.76617740579014070708337155906, −4.70072073473272978855995616794, −3.68892085438331114823756376178, −3.14943036190524285664387660690, −2.3933810831135768765477223338, −1.099876713045097197640564273568, 0.099127686901246320179977654488, 1.30901996674000179711494167935, 2.21672513262107971767319067252, 3.17359010349167560289831457550, 4.22411873422351299569226964149, 4.696485075183646978184981346971, 5.44104471180890313931829254903, 6.467322431946248439096013119795, 7.37115692015561674858854056461, 7.85648151380169217340153511831, 8.698437897017014757696036346187, 9.46505589799225692206270013458, 10.05933780592925232837532843564, 11.05073003656770101671217642523, 11.82008331971796876781871811655, 12.309929620547525782931651834, 13.026951368514345477404688799907, 13.737298277752257948862848543170, 14.70811006861628108327896298866, 15.1996659945877947738295340166, 16.11833397891131072428389372598, 16.51302290013860312753890794931, 17.23169739732496909992469737788, 18.13779771050341902533481703674, 18.69838590383934399162873939333

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