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Properties

Label 1-571-571.117-r0-0-0
Degree $1$
Conductor $571$
Sign $0.999 + 0.0205i$
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  + (−0.986 + 0.164i)2-s + (−0.724 − 0.689i)3-s + (0.945 − 0.324i)4-s + (0.863 + 0.504i)5-s + (0.828 + 0.560i)6-s + (−0.973 + 0.229i)7-s + (−0.879 + 0.475i)8-s + (0.0495 + 0.998i)9-s + (−0.934 − 0.355i)10-s + (−0.934 + 0.355i)11-s + (−0.909 − 0.416i)12-s + (−0.768 − 0.639i)13-s + (0.922 − 0.386i)14-s + (−0.277 − 0.960i)15-s + (0.789 − 0.614i)16-s + (−0.999 + 0.0330i)17-s + ⋯
L(s)  = 1  + (−0.986 + 0.164i)2-s + (−0.724 − 0.689i)3-s + (0.945 − 0.324i)4-s + (0.863 + 0.504i)5-s + (0.828 + 0.560i)6-s + (−0.973 + 0.229i)7-s + (−0.879 + 0.475i)8-s + (0.0495 + 0.998i)9-s + (−0.934 − 0.355i)10-s + (−0.934 + 0.355i)11-s + (−0.909 − 0.416i)12-s + (−0.768 − 0.639i)13-s + (0.922 − 0.386i)14-s + (−0.277 − 0.960i)15-s + (0.789 − 0.614i)16-s + (−0.999 + 0.0330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.999 + 0.0205i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.999 + 0.0205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5511496616 + 0.005656169349i\)
\(L(\frac12)\) \(\approx\) \(0.5511496616 + 0.005656169349i\)
\(L(1)\) \(\approx\) \(0.5374249242 + 0.01657700623i\)
\(L(1)\) \(\approx\) \(0.5374249242 + 0.01657700623i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.986 + 0.164i)T \)
3 \( 1 + (-0.724 - 0.689i)T \)
5 \( 1 + (0.863 + 0.504i)T \)
7 \( 1 + (-0.973 + 0.229i)T \)
11 \( 1 + (-0.934 + 0.355i)T \)
13 \( 1 + (-0.768 - 0.639i)T \)
17 \( 1 + (-0.999 + 0.0330i)T \)
19 \( 1 + (0.431 - 0.901i)T \)
23 \( 1 + (0.180 - 0.983i)T \)
29 \( 1 + (0.245 + 0.969i)T \)
31 \( 1 + (0.789 - 0.614i)T \)
37 \( 1 + (0.371 + 0.928i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (0.965 - 0.261i)T \)
47 \( 1 + (0.789 - 0.614i)T \)
53 \( 1 + (0.701 + 0.712i)T \)
59 \( 1 + (-0.0825 + 0.996i)T \)
61 \( 1 + (-0.461 - 0.887i)T \)
67 \( 1 + (0.701 + 0.712i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.371 + 0.928i)T \)
79 \( 1 + (-0.0165 + 0.999i)T \)
83 \( 1 + (0.828 + 0.560i)T \)
89 \( 1 + (0.828 + 0.560i)T \)
97 \( 1 + (-0.574 + 0.818i)T \)
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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

−23.22401123285577075463018770507, −22.148416234690161199159364375129, −21.41401598777778513469434480849, −20.872262508605352743366395953547, −19.902858914812177644675976162861, −19.039346259750955238759030045004, −17.96527280189117464853185809518, −17.40553672806767604315708960654, −16.53685782353964357301120680237, −16.09479040655585641821405414278, −15.274278996599231909246426267803, −13.808840279279904388466377792816, −12.78507794673037471623907736207, −12.00081617911190386186346829701, −10.95741369884835501281148590888, −10.09532640333802673035336056027, −9.61406989901225013446567210461, −8.89431499404869729829135654722, −7.55280442785042891614196509194, −6.41676676011380763138044377360, −5.81329662933061624355611353484, −4.618417627796693027027514503901, −3.28072270587665949867384428134, −2.157575255966757948028766172320, −0.66384314338426499150775097602, 0.72747132794830731921755486651, 2.477436035133152203402872914663, 2.56672970414509124929704792811, 5.0663915484676126670805738428, 5.86463878844682316362874130665, 6.81720643668333623999159446481, 7.19218561495339015408705009752, 8.47895237737441452125480396830, 9.56851232820216704828227447076, 10.369409744223666279380296222082, 10.895212207299325752261282774519, 12.15541229620054012213957699028, 12.9108223133006680854523543202, 13.758984543403398984166588509464, 15.16880174431859505844938908974, 15.79870357829592838641955501766, 16.91016459313361253788630479662, 17.46569156610887477235640643194, 18.27118091143197055823412393582, 18.71145558566361180023564874049, 19.6991855487392637843909726018, 20.48094163116424497454992292410, 21.85073367479665924846174058652, 22.34330206848546733235247076942, 23.317559953742689013085610009972

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