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Properties

Label 1-571-571.140-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.203 - 0.979i$
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  + (−0.945 − 0.324i)2-s + (0.934 + 0.355i)3-s + (0.789 + 0.614i)4-s + (−0.909 + 0.416i)5-s + (−0.768 − 0.639i)6-s + (−0.701 + 0.712i)7-s + (−0.546 − 0.837i)8-s + (0.746 + 0.665i)9-s + (0.995 − 0.0990i)10-s + (−0.995 − 0.0990i)11-s + (0.518 + 0.854i)12-s + (−0.724 + 0.689i)13-s + (0.894 − 0.446i)14-s + (−0.997 + 0.0660i)15-s + (0.245 + 0.969i)16-s + (0.846 − 0.533i)17-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)2-s + (0.934 + 0.355i)3-s + (0.789 + 0.614i)4-s + (−0.909 + 0.416i)5-s + (−0.768 − 0.639i)6-s + (−0.701 + 0.712i)7-s + (−0.546 − 0.837i)8-s + (0.746 + 0.665i)9-s + (0.995 − 0.0990i)10-s + (−0.995 − 0.0990i)11-s + (0.518 + 0.854i)12-s + (−0.724 + 0.689i)13-s + (0.894 − 0.446i)14-s + (−0.997 + 0.0660i)15-s + (0.245 + 0.969i)16-s + (0.846 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.203 - 0.979i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.203 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003576279245 + 0.004396063764i\)
\(L(\frac12)\) \(\approx\) \(0.003576279245 + 0.004396063764i\)
\(L(1)\) \(\approx\) \(0.5809776399 + 0.1700108072i\)
\(L(1)\) \(\approx\) \(0.5809776399 + 0.1700108072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.945 - 0.324i)T \)
3 \( 1 + (0.934 + 0.355i)T \)
5 \( 1 + (-0.909 + 0.416i)T \)
7 \( 1 + (-0.701 + 0.712i)T \)
11 \( 1 + (-0.995 - 0.0990i)T \)
13 \( 1 + (-0.724 + 0.689i)T \)
17 \( 1 + (0.846 - 0.533i)T \)
19 \( 1 + (-0.965 + 0.261i)T \)
23 \( 1 + (0.0495 + 0.998i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (0.180 + 0.983i)T \)
41 \( 1 + (0.677 + 0.735i)T \)
43 \( 1 + (-0.213 + 0.976i)T \)
47 \( 1 + (-0.245 - 0.969i)T \)
53 \( 1 + (-0.601 - 0.799i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (-0.956 + 0.293i)T \)
67 \( 1 + (-0.601 - 0.799i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.180 - 0.983i)T \)
79 \( 1 + (0.277 - 0.960i)T \)
83 \( 1 + (-0.768 - 0.639i)T \)
89 \( 1 + (0.768 + 0.639i)T \)
97 \( 1 + (0.828 - 0.560i)T \)
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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

−23.653255625987599929050842446, −22.79139480734744059557005827291, −21.12341244791957982265498148366, −20.372315836108578743571669557497, −19.87493304754692819847981518709, −19.044988135456965765097808695675, −18.634281418799463505107572632113, −17.32702656009490403488189377460, −16.60350747540615550288837211167, −15.6517604789810118490237798654, −15.095962461574751721716014881680, −14.20329728782658857290748643743, −12.70396861685123677259803112158, −12.60024400398817676941712049889, −10.91436769515666454260080830348, −10.21000314510862344701467504262, −9.28256111180816206017694303802, −8.3310017874934769592037765773, −7.63798280181334237312525565882, −7.167244515867838492798726930078, −5.88366451528151121101530106142, −4.43143949403445973691366295322, −3.252617298118660712743723963638, −2.293581817924088290533744931232, −0.81283102491874068340027344185, 0.002147608305951028785046030468, 1.89672250419512119615124150170, 2.92274115651769389673193646865, 3.40004192975739948164041008481, 4.75751901483395691780345458049, 6.39562406712357301611948404071, 7.51550127552204865801910255426, 7.97762252333543243073857780951, 9.00395292116962758834082270018, 9.75000136682769509040839459267, 10.50088818757349884485439400807, 11.58049156416004669235040521382, 12.390814175256340518199130568891, 13.31286349894602073274668827056, 14.72837614600121659222556003677, 15.32238402875690404321485426858, 16.085905038713034952337007347685, 16.696163591585711416911632910756, 18.25508197467571137542586215204, 18.840753464022378380852553595289, 19.38484232719471791797861146223, 19.99217033175213070172200780651, 21.20817789650237817609276912968, 21.48937939976280748948510697297, 22.67360371755864062472844194578

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