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Properties

Label 1-571-571.36-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.925 + 0.378i$
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  + (0.789 + 0.614i)2-s + (0.863 − 0.504i)3-s + (0.245 + 0.969i)4-s + (−0.973 + 0.229i)5-s + (0.991 + 0.131i)6-s + (−0.956 − 0.293i)7-s + (−0.401 + 0.915i)8-s + (0.490 − 0.871i)9-s + (−0.909 − 0.416i)10-s + (−0.909 + 0.416i)11-s + (0.701 + 0.712i)12-s + (−0.627 + 0.778i)13-s + (−0.574 − 0.818i)14-s + (−0.724 + 0.689i)15-s + (−0.879 + 0.475i)16-s + (0.180 + 0.983i)17-s + ⋯
L(s)  = 1  + (0.789 + 0.614i)2-s + (0.863 − 0.504i)3-s + (0.245 + 0.969i)4-s + (−0.973 + 0.229i)5-s + (0.991 + 0.131i)6-s + (−0.956 − 0.293i)7-s + (−0.401 + 0.915i)8-s + (0.490 − 0.871i)9-s + (−0.909 − 0.416i)10-s + (−0.909 + 0.416i)11-s + (0.701 + 0.712i)12-s + (−0.627 + 0.778i)13-s + (−0.574 − 0.818i)14-s + (−0.724 + 0.689i)15-s + (−0.879 + 0.475i)16-s + (0.180 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2266053839 + 1.153210869i\)
\(L(\frac12)\) \(\approx\) \(0.2266053839 + 1.153210869i\)
\(L(1)\) \(\approx\) \(1.112837756 + 0.5990225679i\)
\(L(1)\) \(\approx\) \(1.112837756 + 0.5990225679i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.789 + 0.614i)T \)
3 \( 1 + (0.863 - 0.504i)T \)
5 \( 1 + (-0.973 + 0.229i)T \)
7 \( 1 + (-0.956 - 0.293i)T \)
11 \( 1 + (-0.909 + 0.416i)T \)
13 \( 1 + (-0.627 + 0.778i)T \)
17 \( 1 + (0.180 + 0.983i)T \)
19 \( 1 + (-0.995 - 0.0990i)T \)
23 \( 1 + (-0.213 + 0.976i)T \)
29 \( 1 + (0.546 + 0.837i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (0.965 + 0.261i)T \)
41 \( 1 + (-0.0825 + 0.996i)T \)
43 \( 1 + (0.115 - 0.993i)T \)
47 \( 1 + (-0.879 + 0.475i)T \)
53 \( 1 + (-0.340 - 0.940i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (-0.277 - 0.960i)T \)
67 \( 1 + (-0.340 - 0.940i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.965 + 0.261i)T \)
79 \( 1 + (-0.768 + 0.639i)T \)
83 \( 1 + (0.991 + 0.131i)T \)
89 \( 1 + (0.991 + 0.131i)T \)
97 \( 1 + (-0.846 + 0.533i)T \)
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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

−22.73294962489289509586118467206, −22.07716852756794611815868843689, −21.09646496212536495972730650787, −20.42180417978563914158524563271, −19.7117537734987569269878912087, −19.091836477638330941059796072277, −18.41612096477420367150180505455, −16.42096664555771933804145291039, −15.94971440155824661480523890278, −15.149205431929159664246024573733, −14.55178522015208265925269245724, −13.2824023785822253235119262090, −12.85190560978745082300317605876, −11.92999279536489549703702649077, −10.77242845334231572567757061089, −10.1038027271680760930159126963, −9.163238170219450106426094566717, −8.136716288225451536700081675153, −7.16414462914809132573889682190, −5.79540650241740269527083597338, −4.76380314124581273639369261181, −3.96190881361398532334961778708, −2.92400401442380422548305271385, −2.50342604359811604722373480734, −0.37895637433410168695166469723, 2.075095528175933973952658666290, 3.15918700506854793899107895754, 3.80894232004679908324612548249, 4.7828868593392612887992298231, 6.3385747381059517210526057203, 6.99551585433083871043837746454, 7.736758268726758968584998465800, 8.47944837384369415448921042260, 9.63289986343554418954576635223, 10.89106090842061428378516784302, 12.16976940352216474952331515168, 12.714639409290918129517286261, 13.37169443185473242325974793186, 14.49882173291891937198905151005, 15.01933865807677005318368836204, 15.83557967917176645086481264452, 16.57972754501090243599875063690, 17.72725331177631111475121591140, 18.80758291471002339847728393168, 19.597338169103470778714252572545, 20.125226695909811710773207868061, 21.288578904351196835631933011485, 21.96776416018524192318364431205, 23.21837622565027830260154963570, 23.6285342663084257786986857246

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