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Properties

Label 1-571-571.122-r1-0-0
Degree $1$
Conductor $571$
Sign $0.720 - 0.693i$
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  + (0.986 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.535215454 - 2.632357395i\)
\(L(\frac12)\) \(\approx\) \(6.535215454 - 2.632357395i\)
\(L(1)\) \(\approx\) \(3.043903945 - 0.6509860878i\)
\(L(1)\) \(\approx\) \(3.043903945 - 0.6509860878i\)

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

−22.87989068416468236907297001561, −22.25076183927868381252778882063, −21.223853248468735953604225304109, −20.94887059149376856848468547184, −20.30055064804253718519488599602, −19.15493693139426861951728350343, −18.244112050498629653210204913351, −17.07078744354174244289581161071, −16.25515596162906746775093481321, −14.91921875878634278343280617043, −14.79307282445086608870808612188, −14.00945842978300444531813470394, −13.142551017148223393002001599689, −12.240632187992187240063665915436, −10.83787326947404318335553228188, −10.398001966244879122164393642402, −9.70189667338646841516897346998, −8.07387345424560662130933771956, −7.52219682711588870039599353492, −6.113296379800767711545590466850, −5.08625234302412829414096295104, −4.58423892703402385168732184036, −3.17098324094101482285289433477, −2.531260203590503634218967467951, −1.544292949682362250248500383511, 1.154741772432727141003887450602, 2.196306473457290390659241902504, 2.78506515295562478178855399008, 4.28832498778887118715611566930, 5.23409909581020883975812265623, 5.98537243930748374477116526587, 7.19497305154464308232645394157, 7.92261150717392824801992433799, 8.83384588514304438279363370548, 9.97131866026828587559377703557, 11.23202114779967778991572506539, 12.172229414676933452976790067564, 12.86912267891384941062334345057, 13.759842547441032987605260838645, 14.1859059229217666523561476550, 15.05763920509509086393357064715, 15.97581487395706689377071626489, 17.23007105440537847452110039638, 17.702942841288165720442525151319, 18.952086281841652452995501077792, 19.73102041030390329566450817995, 20.93386179456288235690668733532, 21.12072327020603567983254908462, 21.789478621612267953429975559737, 23.313395243797946308409213880873

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