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Properties

Label 1-571-571.101-r1-0-0
Degree $1$
Conductor $571$
Sign $0.742 - 0.669i$
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  + (−0.350 + 0.936i)2-s + (0.381 + 0.924i)3-s + (−0.754 − 0.656i)4-s + (0.202 + 0.979i)5-s + (−0.999 + 0.0330i)6-s + (−0.652 + 0.757i)7-s + (0.879 − 0.475i)8-s + (−0.709 + 0.705i)9-s + (−0.988 − 0.153i)10-s + (−0.360 + 0.932i)11-s + (0.319 − 0.947i)12-s + (0.884 + 0.466i)13-s + (−0.480 − 0.876i)14-s + (−0.828 + 0.560i)15-s + (0.137 + 0.990i)16-s + (−0.984 − 0.175i)17-s + ⋯
L(s)  = 1  + (−0.350 + 0.936i)2-s + (0.381 + 0.924i)3-s + (−0.754 − 0.656i)4-s + (0.202 + 0.979i)5-s + (−0.999 + 0.0330i)6-s + (−0.652 + 0.757i)7-s + (0.879 − 0.475i)8-s + (−0.709 + 0.705i)9-s + (−0.988 − 0.153i)10-s + (−0.360 + 0.932i)11-s + (0.319 − 0.947i)12-s + (0.884 + 0.466i)13-s + (−0.480 − 0.876i)14-s + (−0.828 + 0.560i)15-s + (0.137 + 0.990i)16-s + (−0.984 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.742 - 0.669i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.742 - 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7089497522 + 0.2724192817i\)
\(L(\frac12)\) \(\approx\) \(-0.7089497522 + 0.2724192817i\)
\(L(1)\) \(\approx\) \(0.3103741233 + 0.7219067091i\)
\(L(1)\) \(\approx\) \(0.3103741233 + 0.7219067091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.350 + 0.936i)T \)
3 \( 1 + (0.381 + 0.924i)T \)
5 \( 1 + (0.202 + 0.979i)T \)
7 \( 1 + (-0.652 + 0.757i)T \)
11 \( 1 + (-0.360 + 0.932i)T \)
13 \( 1 + (0.884 + 0.466i)T \)
17 \( 1 + (-0.984 - 0.175i)T \)
19 \( 1 + (0.234 - 0.972i)T \)
23 \( 1 + (0.431 + 0.901i)T \)
29 \( 1 + (-0.962 - 0.272i)T \)
31 \( 1 + (0.789 - 0.614i)T \)
37 \( 1 + (-0.441 + 0.897i)T \)
41 \( 1 + (-0.993 - 0.110i)T \)
43 \( 1 + (0.159 + 0.987i)T \)
47 \( 1 + (0.926 + 0.376i)T \)
53 \( 1 + (-0.999 + 0.0440i)T \)
59 \( 1 + (-0.0825 + 0.996i)T \)
61 \( 1 + (-0.0605 - 0.998i)T \)
67 \( 1 + (0.537 + 0.843i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.556 + 0.831i)T \)
79 \( 1 + (0.421 - 0.906i)T \)
83 \( 1 + (0.528 + 0.849i)T \)
89 \( 1 + (-0.471 + 0.882i)T \)
97 \( 1 + (0.391 - 0.920i)T \)
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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

−22.30675723969974275981620279296, −21.03736875928015738841229662613, −20.470451793736007304468600043278, −19.92314035749260599052956514999, −19.00705028976332716746023397201, −18.40373268911659744023783111691, −17.38813417718568749854595901741, −16.736872750067702409515355951121, −15.81414793496011500403100334400, −14.066604226748684014852966746, −13.50598753913615068831837466752, −12.90662444521785912071547210353, −12.27578246587977780014826477260, −11.10198962480091499655895664746, −10.306679716509789325588123327005, −9.07522716333739964833019448718, −8.545655326748777696077867440281, −7.76015557777002078104694638199, −6.50200215049702336302696985987, −5.399109380646556173307872680157, −3.94974531425361944553202669633, −3.216843866659538045674542978706, −1.95568255497718057191890112892, −0.9413441364189648865659786004, −0.236463838848854156108968864792, 2.057344568137433702106175900115, 3.14753322566155413452082890440, 4.29891022204273488299530600232, 5.31043445012648539848439307334, 6.291995906038654665465122309611, 7.073523339192806649731450898150, 8.193407932344480595681650438853, 9.3362221712439996843605401169, 9.553358620200866778170762372840, 10.66645076047703151481777125657, 11.47240451182859007567384013608, 13.261813674050746468516230917279, 13.73822653497226455066233093990, 14.94150961269969723087230075517, 15.460763793960295208869566210676, 15.784960159804285907170055838416, 17.04848217025573453489833021848, 17.842529937453967117213428462910, 18.714335829723557336781774650159, 19.370756978345538740897889102512, 20.41309434769105815927919732346, 21.55084125543333008105800209221, 22.330803638826493920786080339923, 22.73571951317256104938598553008, 23.742067634520000421376742876603

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