Properties

Label 1-569-569.134-r0-0-0
Degree $1$
Conductor $569$
Sign $0.545 - 0.838i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (0.996 − 0.0883i)2-s + (0.562 − 0.826i)3-s + (0.984 − 0.176i)4-s + (0.633 + 0.773i)5-s + (0.487 − 0.873i)6-s + (−0.787 − 0.616i)7-s + (0.964 − 0.262i)8-s + (−0.367 − 0.930i)9-s + (0.699 + 0.714i)10-s + (0.0663 − 0.997i)11-s + (0.408 − 0.912i)12-s + (0.759 + 0.650i)13-s + (−0.839 − 0.544i)14-s + (0.996 − 0.0883i)15-s + (0.937 − 0.346i)16-s + (−0.598 + 0.801i)17-s + ⋯
L(s)  = 1  + (0.996 − 0.0883i)2-s + (0.562 − 0.826i)3-s + (0.984 − 0.176i)4-s + (0.633 + 0.773i)5-s + (0.487 − 0.873i)6-s + (−0.787 − 0.616i)7-s + (0.964 − 0.262i)8-s + (−0.367 − 0.930i)9-s + (0.699 + 0.714i)10-s + (0.0663 − 0.997i)11-s + (0.408 − 0.912i)12-s + (0.759 + 0.650i)13-s + (−0.839 − 0.544i)14-s + (0.996 − 0.0883i)15-s + (0.937 − 0.346i)16-s + (−0.598 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.934771142 - 1.591781222i\)
\(L(\frac12)\) \(\approx\) \(2.934771142 - 1.591781222i\)
\(L(1)\) \(\approx\) \(2.233104444 - 0.7309175941i\)
\(L(1)\) \(\approx\) \(2.233104444 - 0.7309175941i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.996 - 0.0883i)T \)
3 \( 1 + (0.562 - 0.826i)T \)
5 \( 1 + (0.633 + 0.773i)T \)
7 \( 1 + (-0.787 - 0.616i)T \)
11 \( 1 + (0.0663 - 0.997i)T \)
13 \( 1 + (0.759 + 0.650i)T \)
17 \( 1 + (-0.598 + 0.801i)T \)
19 \( 1 + (0.984 + 0.176i)T \)
23 \( 1 + (-0.952 - 0.304i)T \)
29 \( 1 + (-0.598 - 0.801i)T \)
31 \( 1 + (0.862 + 0.506i)T \)
37 \( 1 + (-0.787 - 0.616i)T \)
41 \( 1 + (-0.952 - 0.304i)T \)
43 \( 1 + (0.996 + 0.0883i)T \)
47 \( 1 + (-0.448 + 0.894i)T \)
53 \( 1 + (0.487 - 0.873i)T \)
59 \( 1 + (-0.448 + 0.894i)T \)
61 \( 1 + (-0.883 + 0.467i)T \)
67 \( 1 + (-0.598 - 0.801i)T \)
71 \( 1 + (0.964 + 0.262i)T \)
73 \( 1 + (0.984 + 0.176i)T \)
79 \( 1 + (-0.525 + 0.850i)T \)
83 \( 1 + (0.240 + 0.970i)T \)
89 \( 1 + (0.862 - 0.506i)T \)
97 \( 1 + (-0.197 + 0.980i)T \)
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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

−23.11081029491449653309979800769, −22.33717201327455983048894192455, −21.90640938311097208759380467503, −20.85082988316854671736509718066, −20.279901862092238361884558561988, −19.86703173892574244311374520294, −18.37580138754199351758706290807, −17.25064322368103870915327351650, −16.21507507210040392512767742118, −15.72019879475941623464749918656, −15.10607947729949839861246144255, −13.84096528860952405031690370622, −13.4403762703551302840451609595, −12.488240209858908752084769199440, −11.6404071608296494875282838737, −10.33130513559226912870744645707, −9.631691407615044853497138752920, −8.77403630812835052954781413063, −7.64865128488916557250881698470, −6.37782907058199372655787945997, −5.40715201618591868856772401028, −4.802315408233219424752701632810, −3.68480240068810754303633641080, −2.77033592224620867454100376357, −1.79922919184378335497450235842, 1.31279868081604108699793635504, 2.35967283231244618264803048164, 3.35074116803886238536389021410, 3.92217765005818526046759400233, 5.82055229093793724500362562615, 6.33039403469130411411795293303, 6.99918279211486276893701944771, 8.06287340736761203027390605554, 9.34142158429366952991919359827, 10.41725977671261875736059172398, 11.249249343159012391111069936274, 12.23578745395675757017395253245, 13.28652323145192567353430518730, 13.81746628994199259110290683135, 14.13325285618534672163967767081, 15.33559711537121282269247561130, 16.2156341266083072490649388628, 17.23574943641584448851606261449, 18.358822858115658291343331325559, 19.18094719496474869885219549067, 19.72072952564922600526385261522, 20.78358792392905710564600178432, 21.4584123718494897343765166450, 22.518492749651519992547552034655, 22.95706067028566871272186263338

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