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Properties

Label 1-571-571.28-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.949 + 0.315i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.926 + 0.376i)2-s + (−0.411 + 0.911i)3-s + (0.716 + 0.697i)4-s + (0.999 + 0.0220i)5-s + (−0.724 + 0.689i)6-s + (−0.601 + 0.799i)7-s + (0.401 + 0.915i)8-s + (−0.660 − 0.750i)9-s + (0.917 + 0.396i)10-s + (0.802 + 0.596i)11-s + (−0.930 + 0.366i)12-s + (0.159 − 0.987i)13-s + (−0.857 + 0.514i)14-s + (−0.431 + 0.901i)15-s + (0.0275 + 0.999i)16-s + (0.609 + 0.792i)17-s + ⋯
L(s)  = 1  + (0.926 + 0.376i)2-s + (−0.411 + 0.911i)3-s + (0.716 + 0.697i)4-s + (0.999 + 0.0220i)5-s + (−0.724 + 0.689i)6-s + (−0.601 + 0.799i)7-s + (0.401 + 0.915i)8-s + (−0.660 − 0.750i)9-s + (0.917 + 0.396i)10-s + (0.802 + 0.596i)11-s + (−0.930 + 0.366i)12-s + (0.159 − 0.987i)13-s + (−0.857 + 0.514i)14-s + (−0.431 + 0.901i)15-s + (0.0275 + 0.999i)16-s + (0.609 + 0.792i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.949 + 0.315i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.949 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6220789641 + 3.846219286i\)
\(L(\frac12)\) \(\approx\) \(0.6220789641 + 3.846219286i\)
\(L(1)\) \(\approx\) \(1.397501765 + 1.368531820i\)
\(L(1)\) \(\approx\) \(1.397501765 + 1.368531820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.926 + 0.376i)T \)
3 \( 1 + (-0.411 + 0.911i)T \)
5 \( 1 + (0.999 + 0.0220i)T \)
7 \( 1 + (-0.601 + 0.799i)T \)
11 \( 1 + (0.802 + 0.596i)T \)
13 \( 1 + (0.159 - 0.987i)T \)
17 \( 1 + (0.609 + 0.792i)T \)
19 \( 1 + (0.739 + 0.673i)T \)
23 \( 1 + (0.746 + 0.665i)T \)
29 \( 1 + (-0.998 - 0.0550i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (-0.889 + 0.456i)T \)
41 \( 1 + (0.821 - 0.569i)T \)
43 \( 1 + (0.509 + 0.860i)T \)
47 \( 1 + (-0.851 + 0.523i)T \)
53 \( 1 + (-0.528 + 0.849i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (0.984 + 0.175i)T \)
67 \( 1 + (-0.471 - 0.882i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.840 - 0.542i)T \)
79 \( 1 + (0.556 - 0.831i)T \)
83 \( 1 + (-0.234 - 0.972i)T \)
89 \( 1 + (-0.959 - 0.282i)T \)
97 \( 1 + (-0.170 - 0.985i)T \)
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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

−22.49966000738444965609249301112, −22.17500208444039153789160425080, −21.05863671146249757208294249769, −20.27486413612737995047083130266, −19.3068809539896492794122617293, −18.753527797377113378712702601623, −17.6479495493345822513880351850, −16.55281577353188515443177257260, −16.31508984869794277549795456514, −14.36048982853808554094253805571, −14.10312410603626582128639794087, −13.267289261615478638842997443006, −12.682225475823927817761849138974, −11.55075408586266111354363007577, −10.99460185180287861870291194924, −9.854921192190321056508124284965, −8.9764888320485513522281337193, −7.02044472161745898710062062339, −6.902089906655429398449622741125, −5.81550030400651092702491610756, −5.05404982301359730660363068170, −3.69229374852729013119830816825, −2.64991133687389923043819816685, −1.51471861127391557775441315285, −0.71852752034503763312599619744, 1.620828545985978625945244530158, 3.014745512536541688295811111263, 3.67991150155037614553909786586, 4.99918785037210043903000312198, 5.792888922866635210018395946181, 6.12075756396345161091259288080, 7.44972369460791272634942865896, 8.87667050907193285194541455117, 9.6432963961493452174620735695, 10.52714108775297047225266199045, 11.58742572571205045614883041775, 12.50254960941174318460918213814, 13.11124675767650216875680069209, 14.43458399522031733092035780470, 14.87996538325423503769146803532, 15.727934250991700975804969948962, 16.60882568003811710081448363978, 17.273410426673558323190944224793, 18.04035973832620606504388051729, 19.477489434320328247188236419457, 20.65071055478494248667297296652, 21.021448996886191846260931556, 22.10994469318713205010473133270, 22.37275791039656598249535552361, 23.01789929473809567813708947195

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