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Properties

Label 1-571-571.492-r0-0-0
Degree $1$
Conductor $571$
Sign $0.0233 + 0.999i$
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  + (0.635 + 0.771i)2-s + (−0.942 − 0.335i)3-s + (−0.191 + 0.981i)4-s + (0.952 − 0.303i)5-s + (−0.340 − 0.940i)6-s + (0.922 + 0.386i)7-s + (−0.879 + 0.475i)8-s + (0.775 + 0.631i)9-s + (0.840 + 0.542i)10-s + (−0.889 − 0.456i)11-s + (0.509 − 0.860i)12-s + (0.618 + 0.785i)13-s + (0.287 + 0.957i)14-s + (−0.999 − 0.0330i)15-s + (−0.926 − 0.376i)16-s + (0.970 − 0.240i)17-s + ⋯
L(s)  = 1  + (0.635 + 0.771i)2-s + (−0.942 − 0.335i)3-s + (−0.191 + 0.981i)4-s + (0.952 − 0.303i)5-s + (−0.340 − 0.940i)6-s + (0.922 + 0.386i)7-s + (−0.879 + 0.475i)8-s + (0.775 + 0.631i)9-s + (0.840 + 0.542i)10-s + (−0.889 − 0.456i)11-s + (0.509 − 0.860i)12-s + (0.618 + 0.785i)13-s + (0.287 + 0.957i)14-s + (−0.999 − 0.0330i)15-s + (−0.926 − 0.376i)16-s + (0.970 − 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0233 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0233 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.0233 + 0.999i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (492, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.0233 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257610660 + 1.228540555i\)
\(L(\frac12)\) \(\approx\) \(1.257610660 + 1.228540555i\)
\(L(1)\) \(\approx\) \(1.203439415 + 0.6168125028i\)
\(L(1)\) \(\approx\) \(1.203439415 + 0.6168125028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.635 + 0.771i)T \)
3 \( 1 + (-0.942 - 0.335i)T \)
5 \( 1 + (0.952 - 0.303i)T \)
7 \( 1 + (0.922 + 0.386i)T \)
11 \( 1 + (-0.889 - 0.456i)T \)
13 \( 1 + (0.618 + 0.785i)T \)
17 \( 1 + (0.970 - 0.240i)T \)
19 \( 1 + (-0.609 + 0.792i)T \)
23 \( 1 + (-0.724 + 0.689i)T \)
29 \( 1 + (0.716 - 0.697i)T \)
31 \( 1 + (0.789 - 0.614i)T \)
37 \( 1 + (0.938 + 0.345i)T \)
41 \( 1 + (-0.592 + 0.805i)T \)
43 \( 1 + (-0.360 - 0.932i)T \)
47 \( 1 + (0.137 + 0.990i)T \)
53 \( 1 + (-0.0605 + 0.998i)T \)
59 \( 1 + (-0.0825 + 0.996i)T \)
61 \( 1 + (-0.782 - 0.622i)T \)
67 \( 1 + (-0.834 - 0.551i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.170 - 0.985i)T \)
79 \( 1 + (0.391 + 0.920i)T \)
83 \( 1 + (0.984 + 0.175i)T \)
89 \( 1 + (-0.644 + 0.764i)T \)
97 \( 1 + (0.731 - 0.681i)T \)
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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

−23.09074763514159653954667108783, −22.06366075820213321383646396752, −21.34096514244819845534840939416, −20.923225431795194079998705980, −20.09273454344920255877209388554, −18.64076613209284802804836201173, −17.96450348536133347704970898459, −17.552453170764406414912522935396, −16.29994082691344942254451429686, −15.22453952237241368057844091815, −14.541216302178907398832894854761, −13.52363077884377851933997477081, −12.78437376715658114917687129289, −11.9186070595827289555561767071, −10.71472962207167779871144342258, −10.56237974681096950679540458587, −9.771621478611144840173979457945, −8.36556634999916840201041845037, −6.88361213647785061206540129965, −5.93371767210421331946554724265, −5.16828140562893816347110905194, −4.51924780067807042074994866423, −3.206317740124846992322749660600, −2.00234096736260672518932000298, −0.93197712325158305302902480234, 1.40068031852982888919250699244, 2.54718523225450557648873035047, 4.245114741211025416885268720382, 5.08528721846623951475489370708, 5.88276787351972275185118871443, 6.29939933628411375830777681407, 7.74596063468824421701815734748, 8.28914189153289216297939529104, 9.61945952628165516700995273941, 10.74925100882163624192190820812, 11.7942354236291547208940237304, 12.36862399311794529428438980025, 13.58612930455962926358941392383, 13.80435014732480990346477323676, 15.048137336348963662004002741685, 16.05786750152227967447078064303, 16.75007083754911718178552710908, 17.40050772207663404714010418280, 18.30734632958791594448164600607, 18.69596180858855359258100337644, 20.7162421748656435402710723774, 21.36665568618581272305286999928, 21.634306795626750330307082833794, 22.8147063393572419267447589177, 23.6975735088470785928324507116

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