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Properties

Label 1-571-571.139-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.284 + 0.958i$
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  + (0.0275 − 0.999i)2-s + (−0.480 + 0.876i)3-s + (−0.998 − 0.0550i)4-s + (0.391 + 0.920i)5-s + (0.863 + 0.504i)6-s + (0.371 + 0.928i)7-s + (−0.0825 + 0.996i)8-s + (−0.537 − 0.843i)9-s + (0.930 − 0.366i)10-s + (−0.782 + 0.622i)11-s + (0.528 − 0.849i)12-s + (0.815 + 0.578i)13-s + (0.938 − 0.345i)14-s + (−0.995 − 0.0990i)15-s + (0.993 + 0.110i)16-s + (0.202 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.0275 − 0.999i)2-s + (−0.480 + 0.876i)3-s + (−0.998 − 0.0550i)4-s + (0.391 + 0.920i)5-s + (0.863 + 0.504i)6-s + (0.371 + 0.928i)7-s + (−0.0825 + 0.996i)8-s + (−0.537 − 0.843i)9-s + (0.930 − 0.366i)10-s + (−0.782 + 0.622i)11-s + (0.528 − 0.849i)12-s + (0.815 + 0.578i)13-s + (0.938 − 0.345i)14-s + (−0.995 − 0.0990i)15-s + (0.993 + 0.110i)16-s + (0.202 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.284 + 0.958i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5456513868 + 0.7308907086i\)
\(L(\frac12)\) \(\approx\) \(0.5456513868 + 0.7308907086i\)
\(L(1)\) \(\approx\) \(0.8278579861 + 0.1995893608i\)
\(L(1)\) \(\approx\) \(0.8278579861 + 0.1995893608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.0275 - 0.999i)T \)
3 \( 1 + (-0.480 + 0.876i)T \)
5 \( 1 + (0.391 + 0.920i)T \)
7 \( 1 + (0.371 + 0.928i)T \)
11 \( 1 + (-0.782 + 0.622i)T \)
13 \( 1 + (0.815 + 0.578i)T \)
17 \( 1 + (0.202 + 0.979i)T \)
19 \( 1 + (-0.126 - 0.991i)T \)
23 \( 1 + (0.652 + 0.757i)T \)
29 \( 1 + (0.975 - 0.218i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (-0.999 - 0.0110i)T \)
41 \( 1 + (-0.754 + 0.656i)T \)
43 \( 1 + (-0.0605 - 0.998i)T \)
47 \( 1 + (-0.592 + 0.805i)T \)
53 \( 1 + (-0.942 + 0.335i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (-0.997 + 0.0770i)T \)
67 \( 1 + (0.761 + 0.648i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.509 - 0.860i)T \)
79 \( 1 + (0.159 + 0.987i)T \)
83 \( 1 + (0.00551 - 0.999i)T \)
89 \( 1 + (-0.868 + 0.495i)T \)
97 \( 1 + (-0.360 + 0.932i)T \)
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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

−23.241866844840099676811359257293, −22.70224358660437729726582095918, −21.3163850621992619395512268738, −20.62539423788264750096860850810, −19.50638404676347674013364914597, −18.327647384644852166606103177656, −18.00687146625095075296334444475, −16.95626007307686515099322187267, −16.49597338815959971054336596097, −15.75771643097353551595416858333, −14.16647958297540356946118659230, −13.761635229334598812677647588012, −12.978944487947617221179383270936, −12.2866408884132980607348103959, −10.89153908220358302620830612046, −10.06596492458134879915395194843, −8.54385537847857396860163370003, −8.20500850379238464866306848718, −7.19641002291768364333648706281, −6.27878632645971840031518356731, −5.323914319292836060496979277157, −4.80543300138605261494150785608, −3.303028480323721086014466828584, −1.42164490839565258941889526763, −0.53644932652940902817158562388, 1.714690488866150893664391825065, 2.72697146469627063643120943350, 3.63234516464832007866910736272, 4.780723456974706796734292158897, 5.549067986744714345822016868419, 6.52924771488177414563475264449, 8.168036040715514236313337588984, 9.16005034293717466943239888574, 9.85949324402008761161978763383, 10.816482328869744473474343599752, 11.22259929508538007946622876326, 12.152228457464601956595931177796, 13.18516017698697574471392458592, 14.19737388586877897943922409766, 15.13830156907534190806293662632, 15.59002790739885375212655287565, 17.19672822700785446698811083490, 17.72925515599869463631203615792, 18.52340288638802960317295053359, 19.22560793811031523325243294731, 20.48210626204673755672240526863, 21.315663155055675142775464510494, 21.5907714529861635418109677516, 22.3719510305953659504587688625, 23.241044134024955581778078565287

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