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Properties

Label 1-2280-2280.2213-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.873 + 0.487i$
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 + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (−0.642 − 0.766i)23-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s − i·37-s + (−0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.342 − 0.939i)47-s + (0.5 − 0.866i)49-s + (0.642 + 0.766i)53-s + (0.939 − 0.342i)59-s + (−0.766 + 0.642i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (−0.642 − 0.766i)23-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s − i·37-s + (−0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.342 − 0.939i)47-s + (0.5 − 0.866i)49-s + (0.642 + 0.766i)53-s + (0.939 − 0.342i)59-s + (−0.766 + 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.873 + 0.487i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (2213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.873 + 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.253276076 + 0.3263152367i\)
\(L(\frac12)\) \(\approx\) \(1.253276076 + 0.3263152367i\)
\(L(1)\) \(\approx\) \(0.9588250866 + 0.08189975668i\)
\(L(1)\) \(\approx\) \(0.9588250866 + 0.08189975668i\)

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 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.342 + 0.939i)T \)
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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

−19.3632815266870276997725532026, −19.184806520084037876276432971086, −18.08350261831579207633103553892, −17.53879450715894569283558303923, −16.573881112713403796678330537600, −15.94413606271646891316370811795, −15.61681983570804597663462923282, −14.36036218746153919417922907393, −13.76804241638284592494628700405, −13.11429305469342697055836307191, −12.50611325060606651431650797285, −11.46824738013584250323722382800, −10.71761594650263903741883571705, −10.24344543983062402855720613925, −9.21679744214548165317132943706, −8.55727936644676717571125169466, −7.77077352012345711517848347177, −6.84067955034090899852065643068, −6.08386190446067350332916750117, −5.546275464154853860771390754086, −4.211751483581049061269422947955, −3.64890999929016181790730854577, −2.85470910459524754388684079514, −1.69823687037251001709556808985, −0.616914962235496657113832038069, 0.752079805319857245480374021095, 2.1170024273260553428788140409, 2.7639604519757349923755469185, 3.706335120992979741059021238381, 4.618500076380478209427990772, 5.471363560047208146124963407260, 6.32904422687776608569878451516, 6.92687991189305786704997709333, 7.900134111715015682899144771708, 8.6987584167205012393244946982, 9.445029755886888740323490352644, 10.1495871979600636426909770271, 10.8501677850423867262781016450, 11.90349587642940579755922656241, 12.374170448171507577734732007678, 13.300951163083296272369392250, 13.694181354974639047056521895541, 14.855557891733786457200566831498, 15.46237883119169633393130536448, 16.07871376394419794949695590667, 16.676201984920711333657349645301, 17.75900577990151276298628732894, 18.395413116320736619336986159826, 18.74552192523027483411406485673, 19.93205624447183811334582179050

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