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Properties

Label 1-571-571.192-r1-0-0
Degree $1$
Conductor $571$
Sign $0.758 + 0.651i$
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  + (−0.975 − 0.218i)2-s + (−0.0715 − 0.997i)3-s + (0.904 + 0.426i)4-s + (−0.381 − 0.924i)5-s + (−0.148 + 0.988i)6-s + (−0.746 − 0.665i)7-s + (−0.789 − 0.614i)8-s + (−0.989 + 0.142i)9-s + (0.170 + 0.985i)10-s + (0.938 − 0.345i)11-s + (0.360 − 0.932i)12-s + (0.996 − 0.0880i)13-s + (0.583 + 0.812i)14-s + (−0.894 + 0.446i)15-s + (0.635 + 0.771i)16-s + (0.968 + 0.250i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.218i)2-s + (−0.0715 − 0.997i)3-s + (0.904 + 0.426i)4-s + (−0.381 − 0.924i)5-s + (−0.148 + 0.988i)6-s + (−0.746 − 0.665i)7-s + (−0.789 − 0.614i)8-s + (−0.989 + 0.142i)9-s + (0.170 + 0.985i)10-s + (0.938 − 0.345i)11-s + (0.360 − 0.932i)12-s + (0.996 − 0.0880i)13-s + (0.583 + 0.812i)14-s + (−0.894 + 0.446i)15-s + (0.635 + 0.771i)16-s + (0.968 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.758 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03181752882 + 0.01179866930i\)
\(L(\frac12)\) \(\approx\) \(0.03181752882 + 0.01179866930i\)
\(L(1)\) \(\approx\) \(0.4151244716 - 0.3354516389i\)
\(L(1)\) \(\approx\) \(0.4151244716 - 0.3354516389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.975 - 0.218i)T \)
3 \( 1 + (-0.0715 - 0.997i)T \)
5 \( 1 + (-0.381 - 0.924i)T \)
7 \( 1 + (-0.746 - 0.665i)T \)
11 \( 1 + (0.938 - 0.345i)T \)
13 \( 1 + (0.996 - 0.0880i)T \)
17 \( 1 + (0.968 + 0.250i)T \)
19 \( 1 + (-0.970 - 0.240i)T \)
23 \( 1 + (-0.340 - 0.940i)T \)
29 \( 1 + (-0.191 - 0.981i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (0.224 + 0.974i)T \)
41 \( 1 + (-0.851 - 0.523i)T \)
43 \( 1 + (-0.441 + 0.897i)T \)
47 \( 1 + (-0.350 + 0.936i)T \)
53 \( 1 + (0.660 - 0.750i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (-0.999 - 0.0110i)T \)
67 \( 1 + (0.319 + 0.947i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.731 + 0.681i)T \)
79 \( 1 + (0.795 - 0.605i)T \)
83 \( 1 + (-0.782 - 0.622i)T \)
89 \( 1 + (-0.930 + 0.366i)T \)
97 \( 1 + (-0.480 - 0.876i)T \)
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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

−23.05911506153594249526605221954, −21.95751368871256601667887535759, −21.38961047109188074070159782659, −20.19045310624951595461799732211, −19.58573654875590033857291492423, −18.69258305423293420902690574977, −18.06517797306099694349251153861, −16.85453745054485997574917350983, −16.31294221850391191603320587876, −15.39248869825172062796321430850, −14.93119672575429071127242356738, −14.0606385262667926701305822865, −12.2817254251149337881411447740, −11.50249726591841241486089388013, −10.73226120478629119691488365347, −9.91420916305833684952352609197, −9.18381561655241361789156551038, −8.40070153626423068970200314927, −7.16080723571377963027552688719, −6.25983472677323916037952727556, −5.55750345364188098940174768525, −3.77634727643961489520623436857, −3.19652763893053030287146445010, −1.84630822327896635991008937550, −0.014500370790157661027588704381, 0.90358099295688424308612814833, 1.62234994386480710706118112639, 3.14595733354928908932394732968, 4.07655655288055041792344647082, 5.962406309698258812045312595108, 6.53300382605348443925741735476, 7.5954900404205557676063062624, 8.40687101390857014450131250096, 9.01111559944684921491201931735, 10.16933977220810540270891514982, 11.22113904773208136089560678302, 11.98950780952089378529920964255, 12.78870989326537467258851041128, 13.43502150226536368696085790616, 14.7187079391599388757178813835, 16.01771089762672618033153865158, 16.83334523348304766998888098595, 17.024222798248620776754567719317, 18.27443845824811989334971378914, 19.095028365868805814224494383907, 19.60567197035447413995593432858, 20.3175450206187681831402869423, 21.077834100981030703705514727394, 22.42817678280269190774351755581, 23.4274809817549362032985882108

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