Properties

Label 1-571-571.114-r1-0-0
Degree $1$
Conductor $571$
Sign $0.479 + 0.877i$
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  + (0.998 + 0.0550i)2-s + (0.0605 − 0.998i)3-s + (0.993 + 0.110i)4-s + (−0.899 − 0.436i)5-s + (0.115 − 0.993i)6-s + (−0.991 + 0.131i)7-s + (0.986 + 0.164i)8-s + (−0.992 − 0.120i)9-s + (−0.874 − 0.485i)10-s + (−0.857 − 0.514i)11-s + (0.170 − 0.985i)12-s + (−0.795 + 0.605i)13-s + (−0.997 + 0.0770i)14-s + (−0.490 + 0.871i)15-s + (0.975 + 0.218i)16-s + (−0.0935 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0550i)2-s + (0.0605 − 0.998i)3-s + (0.993 + 0.110i)4-s + (−0.899 − 0.436i)5-s + (0.115 − 0.993i)6-s + (−0.991 + 0.131i)7-s + (0.986 + 0.164i)8-s + (−0.992 − 0.120i)9-s + (−0.874 − 0.485i)10-s + (−0.857 − 0.514i)11-s + (0.170 − 0.985i)12-s + (−0.795 + 0.605i)13-s + (−0.997 + 0.0770i)14-s + (−0.490 + 0.871i)15-s + (0.975 + 0.218i)16-s + (−0.0935 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.479 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010418581 + 0.5993255376i\)
\(L(\frac12)\) \(\approx\) \(1.010418581 + 0.5993255376i\)
\(L(1)\) \(\approx\) \(1.222023205 - 0.3159739553i\)
\(L(1)\) \(\approx\) \(1.222023205 - 0.3159739553i\)

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 \( 1 + (0.0605 - 0.998i)T \)
5 \( 1 + (-0.899 - 0.436i)T \)
7 \( 1 + (-0.991 + 0.131i)T \)
11 \( 1 + (-0.857 - 0.514i)T \)
13 \( 1 + (-0.795 + 0.605i)T \)
17 \( 1 + (-0.0935 - 0.995i)T \)
19 \( 1 + (0.537 + 0.843i)T \)
23 \( 1 + (0.701 - 0.712i)T \)
29 \( 1 + (0.904 - 0.426i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (0.287 + 0.957i)T \)
41 \( 1 + (-0.137 + 0.990i)T \)
43 \( 1 + (0.731 - 0.681i)T \)
47 \( 1 + (0.298 + 0.954i)T \)
53 \( 1 + (0.360 + 0.932i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.889 - 0.456i)T \)
67 \( 1 + (-0.988 - 0.153i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.685 + 0.728i)T \)
79 \( 1 + (-0.952 - 0.303i)T \)
83 \( 1 + (0.802 + 0.596i)T \)
89 \( 1 + (0.917 - 0.396i)T \)
97 \( 1 + (0.411 - 0.911i)T \)
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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

−22.867788942435988761262852968209, −22.12168183622827977212314463381, −21.51196782494754902323421758266, −20.41161058837474510372006532918, −19.739777284547472427323952416, −19.2976405797811943696173459696, −17.70481417294705455427951440284, −16.58097608747149966680514868323, −15.84615834991566854216524554795, −15.230956593705048396301405700902, −14.79001902983387776323677516066, −13.536262576771176092574446703744, −12.68023510998543087319426066457, −11.861940918418967158964086759282, −10.71620202244344744105175936894, −10.39345155129877343318283088502, −9.223318273571201153940093404, −7.766434434078162369323045074335, −7.085856211348546728181564469561, −5.84668502467907112728484794485, −4.94552531285814647043735332393, −4.03728702457810022698191423126, −3.18891741912008439093797321276, −2.55758674409413688582049998505, −0.21324284342771643183478989150, 1.06424715690933801149384564869, 2.67173054699744328958932250885, 3.1449639707260323467278180174, 4.52892492520707496488589813342, 5.4703985760340675916666117529, 6.507841807296790371890024088943, 7.29679917608412153017226405654, 8.00097713464183327878841742289, 9.17060110327418877445731750883, 10.59948075252787605513520926499, 11.708745084844706485246598001893, 12.230889904817261557077740472317, 12.90440067922659487986943121750, 13.70401959370665216166945259573, 14.52329126667358086986995926168, 15.65309624903660259087449751965, 16.29150732785036848018478904915, 16.98707843041609633687744624276, 18.59430895046088613304767857807, 19.05393241525921092039541642601, 19.97515687603204993036831400011, 20.51432948850003488154010654734, 21.67673912621222529020505549216, 22.72535006200677570571943833934, 23.17013186039664945081710370251

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