Properties

Label 1-2700-2700.1247-r1-0-0
Degree $1$
Conductor $2700$
Sign $0.246 + 0.969i$
Analytic cond. $290.155$
Root an. cond. $290.155$
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  + (0.342 + 0.939i)7-s + (−0.241 − 0.970i)11-s + (−0.694 + 0.719i)13-s + (0.207 + 0.978i)17-s + (0.669 + 0.743i)19-s + (0.829 − 0.559i)23-s + (0.990 − 0.139i)29-s + (0.615 − 0.788i)31-s + (−0.406 + 0.913i)37-s + (0.719 + 0.694i)41-s + (−0.642 + 0.766i)43-s + (0.788 − 0.615i)47-s + (−0.766 + 0.642i)49-s + (−0.951 + 0.309i)53-s + (0.241 − 0.970i)59-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)7-s + (−0.241 − 0.970i)11-s + (−0.694 + 0.719i)13-s + (0.207 + 0.978i)17-s + (0.669 + 0.743i)19-s + (0.829 − 0.559i)23-s + (0.990 − 0.139i)29-s + (0.615 − 0.788i)31-s + (−0.406 + 0.913i)37-s + (0.719 + 0.694i)41-s + (−0.642 + 0.766i)43-s + (0.788 − 0.615i)47-s + (−0.766 + 0.642i)49-s + (−0.951 + 0.309i)53-s + (0.241 − 0.970i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(290.155\)
Root analytic conductor: \(290.155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2700,\ (1:\ ),\ 0.246 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.842477174 + 1.432608607i\)
\(L(\frac12)\) \(\approx\) \(1.842477174 + 1.432608607i\)
\(L(1)\) \(\approx\) \(1.132366922 + 0.2194192532i\)
\(L(1)\) \(\approx\) \(1.132366922 + 0.2194192532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.342 + 0.939i)T \)
11 \( 1 + (-0.241 - 0.970i)T \)
13 \( 1 + (-0.694 + 0.719i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.829 - 0.559i)T \)
29 \( 1 + (0.990 - 0.139i)T \)
31 \( 1 + (0.615 - 0.788i)T \)
37 \( 1 + (-0.406 + 0.913i)T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.788 - 0.615i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (0.139 - 0.990i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (-0.469 + 0.882i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.927 - 0.374i)T \)
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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

−19.1069571611395873369658587497, −17.93174147987685435109033601889, −17.62920284078914825025346033580, −17.081824878022954940200258435965, −15.95063811388877577899180822858, −15.59483367213025855599040411549, −14.59055626632176650856605453672, −14.05316348627877073576694168016, −13.3065615033647210001946619576, −12.54371280055598896762991152194, −11.845732521129768495319224479856, −11.00521235857827744844403888762, −10.270466828753598330848869102, −9.720201648466648700283154565993, −8.869708955549887793746719468116, −7.83508825778278586951912228326, −7.203390778311611451793602406204, −6.86886270891871968564832402511, −5.344051531154999051243211330056, −4.99831909821176792294459794083, −4.14831932346933093758976543945, −3.098134677516174622148399967191, −2.40660908238337236011688647025, −1.18175284775694304371924901207, −0.49092644201548232137107592505, 0.82174461068737375428191877009, 1.805866505176422132815904519081, 2.690499441896786709098622428, 3.42015204673128513561712472217, 4.545491528402658583814200909314, 5.19670403013111836584309889448, 6.08449130754176539391898576294, 6.60410563187352349905201727040, 7.90267945246306772915529966069, 8.24740584812060103265869309606, 9.11502797112336493813758809143, 9.83402920011680442399074686462, 10.68887049912141301454138311830, 11.4960484893112595798736568833, 12.06041737342100699838282501898, 12.74767895666081816510476345230, 13.6585016568503716428150718147, 14.36837101198045708794874057511, 14.973472137224335020085443673169, 15.70236072422458941982098540535, 16.51359981521746915817046628599, 17.053098871439571937526845605696, 17.913463776343942883504242877690, 18.81414983962964068056630184199, 18.98068586353784889340642572966

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