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Properties

Label 1-555-555.89-r0-0-0
Degree $1$
Conductor $555$
Sign $-0.991 + 0.133i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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)2-s + (−0.173 − 0.984i)4-s + (0.939 + 0.342i)7-s + (0.866 + 0.5i)8-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (−0.984 − 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.342 − 0.939i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.173 − 0.984i)28-s + (0.866 + 0.5i)29-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.939 + 0.342i)7-s + (0.866 + 0.5i)8-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (−0.984 − 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.342 − 0.939i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.173 − 0.984i)28-s + (0.866 + 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.991 + 0.133i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ -0.991 + 0.133i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03318113193 + 0.4938365417i\)
\(L(\frac12)\) \(\approx\) \(0.03318113193 + 0.4938365417i\)
\(L(1)\) \(\approx\) \(0.5649962414 + 0.3153949907i\)
\(L(1)\) \(\approx\) \(0.5649962414 + 0.3153949907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.342 + 0.939i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−22.667260398575856271078656049364, −21.89502392644949094719514241697, −21.10590027940995924339650649486, −20.46564560977873933700509045202, −19.56899416072686949378442834369, −18.821764440632804323656918899442, −17.914176242731003021584782897342, −17.26326780406991736391145744435, −16.49023177277082040187922123660, −15.41038903257609243438066242735, −14.25195992881336675913535476077, −13.47085637127999499254335577490, −12.4422655540632287020934089254, −11.64551768101613146127862478128, −10.72183817474416592085490642475, −10.21273346446133530250227961319, −8.96714244138383803709848801518, −8.139050785346024665774519788181, −7.54020561516818705508396901996, −6.172643956937577288334944021423, −4.766798382076345234237876453300, −3.985616548674523571940859697066, −2.63852763545668536954384866460, −1.8136031356616610714571732888, −0.305426154148900606973566994090, 1.61518101834936157633679530016, 2.49693101011443415914824686862, 4.716956537727053475282315124328, 4.83802349580372732582200242438, 6.27171725358405033160138894458, 7.18211763383472226016469313300, 7.97718430474738882132051208142, 8.84672620221624379774833440615, 9.73941006415621122127012621776, 10.65006611441743101781284519077, 11.55357623450526216812433689447, 12.65598149830942992948775335626, 13.82529727914517602176647622733, 14.67398298294802476392146449059, 15.311452614619088377641074121357, 16.070358539822530882446719382180, 17.27055828422616471555328903898, 17.78293725759385082666791626796, 18.34353595536855984373319172285, 19.6555468145687579791842090417, 20.013839370045446154356474311808, 21.31377644077821544548366234816, 22.07979988073067402746323756162, 23.24606495215855886668751062713, 23.86841698182652847418597749120

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