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Properties

Label 1-571-571.121-r0-0-0
Degree $1$
Conductor $571$
Sign $0.896 - 0.443i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (−0.998 + 0.0550i)2-s + (0.930 − 0.366i)3-s + (0.993 − 0.110i)4-s + (0.471 − 0.882i)5-s + (−0.909 + 0.416i)6-s + (0.180 + 0.983i)7-s + (−0.986 + 0.164i)8-s + (0.731 − 0.681i)9-s + (−0.421 + 0.906i)10-s + (0.996 + 0.0880i)11-s + (0.884 − 0.466i)12-s + (0.999 + 0.0220i)13-s + (−0.234 − 0.972i)14-s + (0.115 − 0.993i)15-s + (0.975 − 0.218i)16-s + (−0.660 + 0.750i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0550i)2-s + (0.930 − 0.366i)3-s + (0.993 − 0.110i)4-s + (0.471 − 0.882i)5-s + (−0.909 + 0.416i)6-s + (0.180 + 0.983i)7-s + (−0.986 + 0.164i)8-s + (0.731 − 0.681i)9-s + (−0.421 + 0.906i)10-s + (0.996 + 0.0880i)11-s + (0.884 − 0.466i)12-s + (0.999 + 0.0220i)13-s + (−0.234 − 0.972i)14-s + (0.115 − 0.993i)15-s + (0.975 − 0.218i)16-s + (−0.660 + 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.896 - 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504392373 - 0.3518997236i\)
\(L(\frac12)\) \(\approx\) \(1.504392373 - 0.3518997236i\)
\(L(1)\) \(\approx\) \(1.140693360 - 0.1709861242i\)
\(L(1)\) \(\approx\) \(1.140693360 - 0.1709861242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.998 + 0.0550i)T \)
3 \( 1 + (0.930 - 0.366i)T \)
5 \( 1 + (0.471 - 0.882i)T \)
7 \( 1 + (0.180 + 0.983i)T \)
11 \( 1 + (0.996 + 0.0880i)T \)
13 \( 1 + (0.999 + 0.0220i)T \)
17 \( 1 + (-0.660 + 0.750i)T \)
19 \( 1 + (-0.0605 - 0.998i)T \)
23 \( 1 + (-0.461 + 0.887i)T \)
29 \( 1 + (0.904 + 0.426i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (0.329 + 0.944i)T \)
41 \( 1 + (0.137 + 0.990i)T \)
43 \( 1 + (0.874 - 0.485i)T \)
47 \( 1 + (-0.298 + 0.954i)T \)
53 \( 1 + (0.840 - 0.542i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (-0.709 - 0.705i)T \)
67 \( 1 + (-0.889 - 0.456i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.982 - 0.186i)T \)
79 \( 1 + (0.583 + 0.812i)T \)
83 \( 1 + (0.815 + 0.578i)T \)
89 \( 1 + (0.0935 - 0.995i)T \)
97 \( 1 + (-0.868 - 0.495i)T \)
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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

−23.30765176966168474392849514832, −22.32286205647759880567519990046, −21.31470789341513496648054555183, −20.64394328958825668236934424213, −19.92888256268764784743651317090, −19.19619174897615358749093338958, −18.306496500556680387494479137983, −17.66163257544445895525457317679, −16.52524231461655022888166402210, −15.95267022514111779547587438364, −14.75017881420629685753186447875, −14.20572396766862472784282305647, −13.41213770576005262552110402076, −11.946177353873040734382788190963, −10.68341446137205093667399943598, −10.519583999696052301514969822098, −9.409036444343927518507960399152, −8.7167044954064961793306437509, −7.67223262610719117428460583997, −6.91827203870662569380745466021, −6.064473045122695354632576630569, −4.15961952148062514813587632651, −3.38277978790819955753692678606, −2.27597187019346904994755073290, −1.29288142919482499662466725904, 1.271808286736593938410622268025, 1.86385920888348149340634615792, 2.98182842314640238899604478390, 4.33767194023445828517018263336, 5.913588422759221892128746128620, 6.53132241770665952294495145583, 7.83779513536432628550987925602, 8.72060452318977413813520251087, 9.01398139705284240842819304614, 9.78365063142634268823750242511, 11.18451545891496806716992679276, 12.06940598642699228018861339297, 12.925430819286866395794101384978, 13.85713042077311072292249814317, 15.01081351022076024388561157295, 15.58559493348822201882045097905, 16.52449981862484316602754986674, 17.68825795367223650085460515545, 18.01169335010309755533500107980, 19.14278499663313500440080507058, 19.76809751970256416275547865231, 20.425185939029537868125125776557, 21.33936665309540798178731292918, 21.891507004290879637307054377761, 23.819411148362107092695173087439

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