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Properties

Label 1-285-285.233-r0-0-0
Degree $1$
Conductor $285$
Sign $-0.350 + 0.936i$
Analytic cond. $1.32353$
Root an. cond. $1.32353$
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 + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s + (−0.342 + 0.939i)22-s + (−0.984 − 0.173i)23-s + (0.5 − 0.866i)26-s + (−0.642 + 0.766i)28-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.342 − 0.939i)32-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s + (−0.342 + 0.939i)22-s + (−0.984 − 0.173i)23-s + (0.5 − 0.866i)26-s + (−0.642 + 0.766i)28-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.342 − 0.939i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ -0.350 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.006985038 + 1.451982645i\)
\(L(\frac12)\) \(\approx\) \(1.006985038 + 1.451982645i\)
\(L(1)\) \(\approx\) \(1.216284056 + 0.8595235506i\)
\(L(1)\) \(\approx\) \(1.216284056 + 0.8595235506i\)

Euler product

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

−24.928877154393165531256892927861, −24.10582112272183443254221881239, −23.539031021413712646653347564083, −22.41140340105775940217045277678, −21.57814189251757967086078941309, −20.87312794895679036857213584488, −19.975150790588840680879869077052, −19.06763748747216165555142100557, −18.23437846786161253950964185946, −17.04005779125863315438990075100, −16.01883235146439045540956111402, −14.652145161302275208739975477042, −14.09180860903557564484677322907, −13.34119956660508226622829232662, −11.7367619169198108398520530619, −11.632928068551096201039568711286, −10.34009199892895947790508939994, −9.4082549828519552890667799404, −8.17427634756349020632168028677, −6.8090137370032297771927459115, −5.63813689626552685997232646827, −4.54922307292407308003867741384, −3.67206768191602506571804355062, −2.28022520360641042496238572088, −1.05864125903693536753209181268, 1.90643970653474934009376421231, 3.32875209837956301408737183074, 4.57350474854544794977972044799, 5.40248829943423198438817899341, 6.47115640783248516516074035861, 7.698367165466892705726827595121, 8.33839172122044064189983256630, 9.60435727722094108330366751038, 10.97143927198147406746721618339, 12.295015864481643410800207580390, 12.59235019603937237084072755975, 14.17098816969470849121993067582, 14.680856171611738599942989537353, 15.51070383253580843522958092991, 16.55425111368502244337214872586, 17.72932635147078787965784913577, 17.96105273912092515428711402276, 19.62320243180651074473402034366, 20.62022825563378234132989317595, 21.54854678536266956983404641354, 22.26873643103563977010988760222, 23.24136291035967510326599515121, 24.02265537393960881124450138810, 25.0172104915809406990376608199, 25.42681316327898646565794046526

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