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Properties

Label 1-571-571.165-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.577 - 0.816i$
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 + (0.234 + 0.972i)3-s + (−0.754 + 0.656i)4-s + (−0.868 − 0.495i)5-s + (0.828 − 0.560i)6-s + (0.973 + 0.229i)7-s + (0.879 + 0.475i)8-s + (−0.889 + 0.456i)9-s + (−0.159 + 0.987i)10-s + (0.775 − 0.631i)11-s + (−0.815 − 0.578i)12-s + (−0.170 − 0.985i)13-s + (−0.126 − 0.991i)14-s + (0.277 − 0.960i)15-s + (0.137 − 0.990i)16-s + (−0.471 − 0.882i)17-s + ⋯
L(s)  = 1  + (−0.350 − 0.936i)2-s + (0.234 + 0.972i)3-s + (−0.754 + 0.656i)4-s + (−0.868 − 0.495i)5-s + (0.828 − 0.560i)6-s + (0.973 + 0.229i)7-s + (0.879 + 0.475i)8-s + (−0.889 + 0.456i)9-s + (−0.159 + 0.987i)10-s + (0.775 − 0.631i)11-s + (−0.815 − 0.578i)12-s + (−0.170 − 0.985i)13-s + (−0.126 − 0.991i)14-s + (0.277 − 0.960i)15-s + (0.137 − 0.990i)16-s + (−0.471 − 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4574590000 - 0.8840126878i\)
\(L(\frac12)\) \(\approx\) \(0.4574590000 - 0.8840126878i\)
\(L(1)\) \(\approx\) \(0.7794653196 - 0.2379445330i\)
\(L(1)\) \(\approx\) \(0.7794653196 - 0.2379445330i\)

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 \( 1 + (0.234 + 0.972i)T \)
5 \( 1 + (-0.868 - 0.495i)T \)
7 \( 1 + (0.973 + 0.229i)T \)
11 \( 1 + (0.775 - 0.631i)T \)
13 \( 1 + (-0.170 - 0.985i)T \)
17 \( 1 + (-0.471 - 0.882i)T \)
19 \( 1 + (0.997 + 0.0770i)T \)
23 \( 1 + (0.180 + 0.983i)T \)
29 \( 1 + (-0.962 + 0.272i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (-0.993 + 0.110i)T \)
43 \( 1 + (-0.709 + 0.705i)T \)
47 \( 1 + (0.926 - 0.376i)T \)
53 \( 1 + (-0.266 - 0.963i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (-0.537 - 0.843i)T \)
67 \( 1 + (0.968 + 0.250i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.618 - 0.785i)T \)
79 \( 1 + (-0.874 - 0.485i)T \)
83 \( 1 + (0.0715 + 0.997i)T \)
89 \( 1 + (0.899 + 0.436i)T \)
97 \( 1 + (0.996 - 0.0880i)T \)
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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

−23.602162934651326401288387549570, −22.82004235672038731515656396207, −22.07334433799683703842508691408, −20.44468859859920940446505265242, −19.79504094603337390773318036385, −18.82291529148323237202364687821, −18.473065514130778993620402881906, −17.31769828753156187299265128946, −16.97740274463277786367281384165, −15.5734336020874508400712244724, −14.77084253733935759046565568591, −14.29779437273209960672037676896, −13.452021284326187090124082538090, −12.1369448485433228147160734776, −11.50202239637105805103041281153, −10.39446391650345613582494221113, −9.01225554966643791739335383288, −8.38508931608871043721361371714, −7.36933187956388224218753076011, −7.04033602280854417257649709274, −6.05291836628957970678011899811, −4.655772175527077620409073905353, −3.84544257939204379744886139411, −2.083278405558002650130821008467, −1.08281173377569752121850944095, 0.31593848883162737166771308499, 1.53445100166586040330569247955, 3.09968747176068523640782762861, 3.63834430173403691627338869443, 4.84764737852522525153775922016, 5.26365744280361476894040463586, 7.469530204641621173895106285183, 8.29252979746405285374005344382, 8.91981554904155097369997209274, 9.74748762333153753260641113003, 10.891304287061647886314118971525, 11.52347294592551516544834966648, 12.02104864374289319162951800998, 13.41367586600113478513811346500, 14.21521786429134841098299308470, 15.24738224512838446781872124978, 16.03174384890628444769166208592, 16.9896105854064634778776934239, 17.697649582878986477469847813013, 18.790455945316992591682916496949, 19.7447139531968111367315520637, 20.31293011204465500607387204005, 20.83269538143153333795672743855, 21.844796878485748888577797799537, 22.41395570833832078864331175417

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