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Properties

Label 1-571-571.78-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.737 - 0.675i$
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  + (−0.754 − 0.656i)2-s + (−0.889 − 0.456i)3-s + (0.137 + 0.990i)4-s + (0.509 − 0.860i)5-s + (0.371 + 0.928i)6-s + (0.894 − 0.446i)7-s + (0.546 − 0.837i)8-s + (0.583 + 0.812i)9-s + (−0.949 + 0.314i)10-s + (0.202 + 0.979i)11-s + (0.329 − 0.944i)12-s + (−0.942 − 0.335i)13-s + (−0.968 − 0.250i)14-s + (−0.846 + 0.533i)15-s + (−0.962 + 0.272i)16-s + (−0.556 − 0.831i)17-s + ⋯
L(s)  = 1  + (−0.754 − 0.656i)2-s + (−0.889 − 0.456i)3-s + (0.137 + 0.990i)4-s + (0.509 − 0.860i)5-s + (0.371 + 0.928i)6-s + (0.894 − 0.446i)7-s + (0.546 − 0.837i)8-s + (0.583 + 0.812i)9-s + (−0.949 + 0.314i)10-s + (0.202 + 0.979i)11-s + (0.329 − 0.944i)12-s + (−0.942 − 0.335i)13-s + (−0.968 − 0.250i)14-s + (−0.846 + 0.533i)15-s + (−0.962 + 0.272i)16-s + (−0.556 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.737 - 0.675i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.737 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2745800267 - 0.7066365401i\)
\(L(\frac12)\) \(\approx\) \(0.2745800267 - 0.7066365401i\)
\(L(1)\) \(\approx\) \(0.5422571915 - 0.4025421226i\)
\(L(1)\) \(\approx\) \(0.5422571915 - 0.4025421226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.754 - 0.656i)T \)
3 \( 1 + (-0.889 - 0.456i)T \)
5 \( 1 + (0.509 - 0.860i)T \)
7 \( 1 + (0.894 - 0.446i)T \)
11 \( 1 + (0.202 + 0.979i)T \)
13 \( 1 + (-0.942 - 0.335i)T \)
17 \( 1 + (-0.556 - 0.831i)T \)
19 \( 1 + (0.988 - 0.153i)T \)
23 \( 1 + (-0.934 - 0.355i)T \)
29 \( 1 + (0.851 + 0.523i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (0.959 + 0.282i)T \)
41 \( 1 + (0.975 + 0.218i)T \)
43 \( 1 + (0.00551 + 0.999i)T \)
47 \( 1 + (0.716 + 0.697i)T \)
53 \( 1 + (-0.857 - 0.514i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (-0.421 - 0.906i)T \)
67 \( 1 + (0.874 - 0.485i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.234 - 0.972i)T \)
79 \( 1 + (0.528 - 0.849i)T \)
83 \( 1 + (-0.989 - 0.142i)T \)
89 \( 1 + (0.618 - 0.785i)T \)
97 \( 1 + (0.984 + 0.175i)T \)
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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

−23.78162377271153668962563327542, −22.76121595801551689579887319842, −21.71617790230770739169448321017, −21.574880387785149825597473410520, −20.096877665662162594826630688568, −19.03392727726235491638042666688, −18.29065279492824429410021494770, −17.583795184473645820594594466877, −17.12777196126937141770668627913, −16.03098983668118027547796320728, −15.34134796342163583924435829510, −14.43111938565882485165720454731, −13.860949352591239506949087496190, −12.07795863227186259830645171636, −11.304959288158302742958572270200, −10.60954010058713836374974029601, −9.82084807532162381151745718556, −8.935403320849009883221022374821, −7.80793371713462029043170015154, −6.81791796842506769746455018082, −5.92851716030751895598358669309, −5.39471427065346487121037154657, −4.18806943284072243472102916989, −2.488531385709067508519566809284, −1.26467257892685654436490558104, 0.64241145581743564738136257370, 1.60929516596011587285741716465, 2.478002343379954466873774485865, 4.58106790372539718524484882402, 4.76504393656385552787100314129, 6.29904293287618194217808617595, 7.51058834036300544018324398315, 7.91201519879171874758875604880, 9.378509933534207281210098099, 9.9350162231534836569551145513, 10.96286138907669837767932799687, 11.85536882907559278227608114176, 12.382786713215505747367080477235, 13.25686279956208383901076180329, 14.19285127306124152140587325754, 15.7836341231008246771549175591, 16.588347843450247304921137172063, 17.44572884666391404940435638192, 17.74229705949335446688194073893, 18.42936013088211987268710187664, 19.9446778523013749878178509406, 20.14699769904073135536342416219, 21.158748061412292247820294031694, 22.053968121184216784304152675390, 22.70441877979979769331527876082

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