Properties

Label 1-569-569.17-r0-0-0
Degree $1$
Conductor $569$
Sign $0.285 - 0.958i$
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  + (0.921 − 0.387i)2-s + (0.945 + 0.325i)3-s + (0.699 − 0.714i)4-s + (−0.666 − 0.745i)5-s + (0.997 − 0.0663i)6-s + (−0.154 − 0.988i)7-s + (0.367 − 0.930i)8-s + (0.787 + 0.616i)9-s + (−0.903 − 0.428i)10-s + (0.467 + 0.883i)11-s + (0.894 − 0.448i)12-s + (0.999 + 0.0442i)13-s + (−0.525 − 0.850i)14-s + (−0.387 − 0.921i)15-s + (−0.0221 − 0.999i)16-s + (−0.862 − 0.506i)17-s + ⋯
L(s)  = 1  + (0.921 − 0.387i)2-s + (0.945 + 0.325i)3-s + (0.699 − 0.714i)4-s + (−0.666 − 0.745i)5-s + (0.997 − 0.0663i)6-s + (−0.154 − 0.988i)7-s + (0.367 − 0.930i)8-s + (0.787 + 0.616i)9-s + (−0.903 − 0.428i)10-s + (0.467 + 0.883i)11-s + (0.894 − 0.448i)12-s + (0.999 + 0.0442i)13-s + (−0.525 − 0.850i)14-s + (−0.387 − 0.921i)15-s + (−0.0221 − 0.999i)16-s + (−0.862 − 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.285 - 0.958i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.285 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.477323823 - 1.847516549i\)
\(L(\frac12)\) \(\approx\) \(2.477323823 - 1.847516549i\)
\(L(1)\) \(\approx\) \(2.042510288 - 0.8387561665i\)
\(L(1)\) \(\approx\) \(2.042510288 - 0.8387561665i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.921 - 0.387i)T \)
3 \( 1 + (0.945 + 0.325i)T \)
5 \( 1 + (-0.666 - 0.745i)T \)
7 \( 1 + (-0.154 - 0.988i)T \)
11 \( 1 + (0.467 + 0.883i)T \)
13 \( 1 + (0.999 + 0.0442i)T \)
17 \( 1 + (-0.862 - 0.506i)T \)
19 \( 1 + (-0.714 + 0.699i)T \)
23 \( 1 + (-0.176 - 0.984i)T \)
29 \( 1 + (-0.506 - 0.862i)T \)
31 \( 1 + (0.683 + 0.730i)T \)
37 \( 1 + (0.988 - 0.154i)T \)
41 \( 1 + (0.984 - 0.176i)T \)
43 \( 1 + (-0.921 - 0.387i)T \)
47 \( 1 + (-0.262 - 0.964i)T \)
53 \( 1 + (-0.997 + 0.0663i)T \)
59 \( 1 + (0.262 + 0.964i)T \)
61 \( 1 + (-0.814 + 0.580i)T \)
67 \( 1 + (-0.862 + 0.506i)T \)
71 \( 1 + (0.367 + 0.930i)T \)
73 \( 1 + (0.714 - 0.699i)T \)
79 \( 1 + (0.991 + 0.132i)T \)
83 \( 1 + (0.304 + 0.952i)T \)
89 \( 1 + (-0.683 + 0.730i)T \)
97 \( 1 + (0.993 + 0.110i)T \)
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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

−23.67616134710080431104693261981, −22.537674727200819141183231999314, −21.76658023897119911948142695031, −21.241104514663236812034450262, −20.0078055227861065580792428031, −19.40572284326331170208018861360, −18.59869988849876199474096829535, −17.681558173219905911665856351460, −16.22719871326410816435222823283, −15.47833410899895943538737942311, −15.04354035279015576285905214843, −14.17980584354590106566798924005, −13.32625802708374424214494325194, −12.64345389595416934881154345689, −11.47459443082691360646578395647, −11.01176818230724483689858890287, −9.24287554733977460096148796507, −8.40406313752279015431414347803, −7.747392824050519914025031846805, −6.43470862039708132108710484396, −6.199456846655106049484918631063, −4.491214981313506214800232830629, −3.519375397010005118841660415956, −2.944596744465678392008846607996, −1.86309696207393632712850694598, 1.15861259738881219755467019136, 2.25771613256647723537611609713, 3.61784976708998722399390693112, 4.20758635725396634089158616410, 4.72175123829347181080584793965, 6.36940178933675037661630426614, 7.29563560318323805747543346432, 8.2780750980867089004325054564, 9.32648447857010552528614173238, 10.26351391789688139014595598854, 11.09786321629473604037564548915, 12.19677362204031653958557858378, 13.06573017045095000235468000684, 13.62459785868715417846613326079, 14.576419262544798888998156563, 15.329023848435697085517436204610, 16.12274969140453540962504951750, 16.81219230256199942146930869811, 18.38300070865142936707135666202, 19.54950327316416154219275351477, 19.87654986559973878065782687038, 20.78120191838028812446969117546, 20.9317286788840951062945468883, 22.3938018288205300380686369532, 23.05082094023826164809567613020

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