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Properties

Label 1-571-571.105-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.987 + 0.159i$
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  + (−0.401 + 0.915i)2-s + (0.601 + 0.799i)3-s + (−0.677 − 0.735i)4-s + (0.371 + 0.928i)5-s + (−0.973 + 0.229i)6-s + (0.965 + 0.261i)7-s + (0.945 − 0.324i)8-s + (−0.277 + 0.960i)9-s + (−0.999 − 0.0330i)10-s + (−0.999 + 0.0330i)11-s + (0.180 − 0.983i)12-s + (0.701 − 0.712i)13-s + (−0.627 + 0.778i)14-s + (−0.518 + 0.854i)15-s + (−0.0825 + 0.996i)16-s + (0.652 − 0.757i)17-s + ⋯
L(s)  = 1  + (−0.401 + 0.915i)2-s + (0.601 + 0.799i)3-s + (−0.677 − 0.735i)4-s + (0.371 + 0.928i)5-s + (−0.973 + 0.229i)6-s + (0.965 + 0.261i)7-s + (0.945 − 0.324i)8-s + (−0.277 + 0.960i)9-s + (−0.999 − 0.0330i)10-s + (−0.999 + 0.0330i)11-s + (0.180 − 0.983i)12-s + (0.701 − 0.712i)13-s + (−0.627 + 0.778i)14-s + (−0.518 + 0.854i)15-s + (−0.0825 + 0.996i)16-s + (0.652 − 0.757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.987 + 0.159i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1123078346 + 1.399272244i\)
\(L(\frac12)\) \(\approx\) \(0.1123078346 + 1.399272244i\)
\(L(1)\) \(\approx\) \(0.6874653189 + 0.8768455874i\)
\(L(1)\) \(\approx\) \(0.6874653189 + 0.8768455874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.401 + 0.915i)T \)
3 \( 1 + (0.601 + 0.799i)T \)
5 \( 1 + (0.371 + 0.928i)T \)
7 \( 1 + (0.965 + 0.261i)T \)
11 \( 1 + (-0.999 + 0.0330i)T \)
13 \( 1 + (0.701 - 0.712i)T \)
17 \( 1 + (0.652 - 0.757i)T \)
19 \( 1 + (-0.574 + 0.818i)T \)
23 \( 1 + (-0.0165 + 0.999i)T \)
29 \( 1 + (-0.986 + 0.164i)T \)
31 \( 1 + (-0.0825 + 0.996i)T \)
37 \( 1 + (0.894 - 0.446i)T \)
41 \( 1 + (0.245 - 0.969i)T \)
43 \( 1 + (0.828 - 0.560i)T \)
47 \( 1 + (-0.0825 + 0.996i)T \)
53 \( 1 + (-0.213 + 0.976i)T \)
59 \( 1 + (0.546 - 0.837i)T \)
61 \( 1 + (-0.995 - 0.0990i)T \)
67 \( 1 + (-0.213 + 0.976i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.894 - 0.446i)T \)
79 \( 1 + (-0.909 - 0.416i)T \)
83 \( 1 + (-0.973 + 0.229i)T \)
89 \( 1 + (-0.973 + 0.229i)T \)
97 \( 1 + (0.980 + 0.197i)T \)
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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

−22.97911185081457934565180959227, −21.30844211422399510031918626383, −21.163525211705433619108784035920, −20.37025426086582634775056657045, −19.65683866139467495929118873842, −18.56612067224283000693468362194, −18.18938865552926940478607855168, −17.169805287338522979989644927229, −16.56263625554635814552988794988, −15.02091679726630853909955676442, −14.03381785661820240334342175820, −13.16790691705860212624834837257, −12.842827522490437697717656987731, −11.72347663776843991440660304374, −10.945229114942261053333614737256, −9.80000794223690302094852922880, −8.82703390088576233818641997134, −8.23668042294080630940583834527, −7.58763428849786384734630531234, −6.068721221313051895647281262076, −4.774807350218517820379955677065, −3.91812763304744095757031100207, −2.475373006046033308805188764043, −1.76411668661264320564763377649, −0.80351191088172630611905944769, 1.7161122360098272137309707636, 2.93536461139389929740614687098, 4.09335582679288052945804368415, 5.41472615589911311264785810675, 5.713392606357444714659197175399, 7.43200825009776373712091259090, 7.84878942821881740422135517918, 8.83303275724388016230807707975, 9.777746464906740471628465206680, 10.59791104900503463525049281766, 11.10930165415573854446161995911, 12.96095927580058499963941829409, 14.09000948663328223805362309002, 14.34903371418407576869692456843, 15.40510755386978280135460116469, 15.72682531270702687220177979078, 16.90407506194320795456507018640, 17.86206381128174657487533765139, 18.460443404635263564444872093102, 19.19003557064647392612837336280, 20.50191249221014727706468955363, 21.12653322201941662820655937832, 22.00968613488609032198694001567, 22.946830581911762145593879010049, 23.57901722563947989941676773458

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