↑

Properties

Label 1-571-571.19-r1-0-0
Degree $1$
Conductor $571$
Sign $0.989 + 0.144i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.962 − 0.272i)2-s + (0.319 − 0.947i)3-s + (0.851 − 0.523i)4-s + (−0.537 + 0.843i)5-s + (0.0495 − 0.998i)6-s + (0.277 + 0.960i)7-s + (0.677 − 0.735i)8-s + (−0.795 − 0.605i)9-s + (−0.287 + 0.957i)10-s + (0.685 − 0.728i)11-s + (−0.224 − 0.974i)12-s + (0.202 + 0.979i)13-s + (0.528 + 0.849i)14-s + (0.627 + 0.778i)15-s + (0.451 − 0.892i)16-s + (0.709 + 0.705i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.272i)2-s + (0.319 − 0.947i)3-s + (0.851 − 0.523i)4-s + (−0.537 + 0.843i)5-s + (0.0495 − 0.998i)6-s + (0.277 + 0.960i)7-s + (0.677 − 0.735i)8-s + (−0.795 − 0.605i)9-s + (−0.287 + 0.957i)10-s + (0.685 − 0.728i)11-s + (−0.224 − 0.974i)12-s + (0.202 + 0.979i)13-s + (0.528 + 0.849i)14-s + (0.627 + 0.778i)15-s + (0.451 − 0.892i)16-s + (0.709 + 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.160143846 + 0.3013144820i\)
\(L(\frac12)\) \(\approx\) \(4.160143846 + 0.3013144820i\)
\(L(1)\) \(\approx\) \(2.071338622 - 0.3041496359i\)
\(L(1)\) \(\approx\) \(2.071338622 - 0.3041496359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.962 - 0.272i)T \)
3 \( 1 + (0.319 - 0.947i)T \)
5 \( 1 + (-0.537 + 0.843i)T \)
7 \( 1 + (0.277 + 0.960i)T \)
11 \( 1 + (0.685 - 0.728i)T \)
13 \( 1 + (0.202 + 0.979i)T \)
17 \( 1 + (0.709 + 0.705i)T \)
19 \( 1 + (-0.815 + 0.578i)T \)
23 \( 1 + (0.115 + 0.993i)T \)
29 \( 1 + (-0.592 + 0.805i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.411 + 0.911i)T \)
41 \( 1 + (-0.635 - 0.771i)T \)
43 \( 1 + (0.999 - 0.0220i)T \)
47 \( 1 + (0.998 - 0.0550i)T \)
53 \( 1 + (0.556 - 0.831i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.170 - 0.985i)T \)
67 \( 1 + (0.441 + 0.897i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.583 + 0.812i)T \)
79 \( 1 + (0.609 - 0.792i)T \)
83 \( 1 + (0.840 + 0.542i)T \)
89 \( 1 + (0.889 - 0.456i)T \)
97 \( 1 + (0.761 + 0.648i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.902466769186707066844295178250, −22.44892854050582591650734071135, −21.17825617470316778117362760652, −20.55629734042663941860736438973, −20.20273821774106982989882761055, −19.37232807039597872734295416535, −17.45280087241399295551080974494, −16.85949867585072515659477649949, −16.2170450198902104244493201431, −15.23575916250222946739686717156, −14.76643955017621670904585708457, −13.74788335005651937966880909894, −12.96141806405525823583377560895, −11.98157442192164514143391998719, −11.10298626895766535413820395023, −10.244923396899652693012000939518, −9.09176316308328283324432082474, −8.009658059311601205218380177736, −7.40288686591677159804168534647, −5.99041020809725249639130224298, −4.86900612905844185242443971214, −4.32775983682562784160928839458, −3.64968412789473009678117237265, −2.39355346601660280965272577415, −0.715074635910190384343270411451, 1.31748917355074978076616819996, 2.18954835556911149488132533869, 3.25339278656899453311026257421, 3.93533143564392296438159031261, 5.56323813169641993622728601418, 6.29657032618283791265451303380, 7.0199397817252603378941718627, 8.098024907755902262817161125277, 9.03134172912775371301789454205, 10.50671406175071347719174336605, 11.58882431405614404503773016883, 11.84902816413712886887425986042, 12.77495601445960519854940231263, 13.87945685034525655349232448246, 14.49177001772102113925182142229, 15.01914547738338884651603280356, 16.08264000437064870988499126025, 17.218735974302270411815488710226, 18.548750595724938078533488799194, 19.11123735181198143804712225715, 19.40375380313870450043000527503, 20.68189087793548491654167598087, 21.61030128105283223801339661842, 22.139780951532826496168491546387, 23.28417066349466928981112434646

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