Properties

Label 1-2280-2280.2027-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.850 + 0.525i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  + (−0.866 + 0.5i)7-s + (0.5 − 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.342 + 0.939i)23-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s i·37-s + (0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.984 + 0.173i)47-s + (0.5 − 0.866i)49-s + (0.342 − 0.939i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (0.5 − 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.342 + 0.939i)23-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s i·37-s + (0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.984 + 0.173i)47-s + (0.5 − 0.866i)49-s + (0.342 − 0.939i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.073213917 + 0.3044991084i\)
\(L(\frac12)\) \(\approx\) \(1.073213917 + 0.3044991084i\)
\(L(1)\) \(\approx\) \(0.8977796576 + 0.03302692313i\)
\(L(1)\) \(\approx\) \(0.8977796576 + 0.03302692313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (0.342 - 0.939i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.642 - 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

−19.70234960244698453555517528978, −19.01902612614589039664084899910, −18.12612657032936664964003191045, −17.293991036141641174821386688493, −16.86231202218196497127044345375, −15.94405210654617491085697783414, −15.42348524897971932286425465850, −14.31616032961048616782634787971, −14.03633419017304792725208924960, −12.82144875032310240896494493524, −12.54926229257684307933657377418, −11.61074066226391562094872349751, −10.755589747697609545814579160882, −9.96973213862356771807329169901, −9.376866715711109817294383678896, −8.66128234351106490128193165712, −7.53270195357984700782472724200, −6.81701074665399145676682594543, −6.416494684070642164047453425785, −5.22814220321097110109256336914, −4.23170411415379070663246878539, −3.87114386087311585888434001579, −2.52941764829933433570618899299, −1.930504791281645169687038933, −0.51200643828659134068844156568, 0.74449622165713554621471333567, 2.02988961862194424687015963796, 2.97492349556593247007679551770, 3.54682700272061682683154460962, 4.6494075473226804572027501534, 5.56761100294812523076923278243, 6.22458062891440657015458748102, 6.96351421140718520631147450390, 7.91933088610455166177542825491, 8.717832736529120627926368256573, 9.464215851829335122129529692444, 10.02454008481481246307131260161, 11.11303536604867057422800390474, 11.613683371674261371733092202828, 12.601420905280672715376731160471, 13.11480423416930086717030474615, 13.85807023810149017071733791404, 14.80588217842641599312329096760, 15.38417651130625119590051992584, 16.21388271871593297770317418620, 16.68207810036200632995583524458, 17.71088271597395170951095745117, 18.19439153564256366968571675984, 19.20401421891005920418718801995, 19.63092576267192459995459493144

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