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Properties

Label 1-571-571.7-r1-0-0
Degree $1$
Conductor $571$
Sign $0.194 + 0.980i$
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  + (0.677 − 0.735i)2-s + (−0.828 + 0.560i)3-s + (−0.0825 − 0.996i)4-s + (0.991 + 0.131i)5-s + (−0.148 + 0.988i)6-s + (−0.746 − 0.665i)7-s + (−0.789 − 0.614i)8-s + (0.371 − 0.928i)9-s + (0.768 − 0.639i)10-s + (−0.768 − 0.639i)11-s + (0.627 + 0.778i)12-s + (−0.574 − 0.818i)13-s + (−0.995 + 0.0990i)14-s + (−0.894 + 0.446i)15-s + (−0.986 + 0.164i)16-s + (−0.701 + 0.712i)17-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)2-s + (−0.828 + 0.560i)3-s + (−0.0825 − 0.996i)4-s + (0.991 + 0.131i)5-s + (−0.148 + 0.988i)6-s + (−0.746 − 0.665i)7-s + (−0.789 − 0.614i)8-s + (0.371 − 0.928i)9-s + (0.768 − 0.639i)10-s + (−0.768 − 0.639i)11-s + (0.627 + 0.778i)12-s + (−0.574 − 0.818i)13-s + (−0.995 + 0.0990i)14-s + (−0.894 + 0.446i)15-s + (−0.986 + 0.164i)16-s + (−0.701 + 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.194 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1974883198 + 0.1622126510i\)
\(L(\frac12)\) \(\approx\) \(0.1974883198 + 0.1622126510i\)
\(L(1)\) \(\approx\) \(0.8577763001 - 0.3961963910i\)
\(L(1)\) \(\approx\) \(0.8577763001 - 0.3961963910i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.677 - 0.735i)T \)
3 \( 1 + (-0.828 + 0.560i)T \)
5 \( 1 + (0.991 + 0.131i)T \)
7 \( 1 + (-0.746 - 0.665i)T \)
11 \( 1 + (-0.768 - 0.639i)T \)
13 \( 1 + (-0.574 - 0.818i)T \)
17 \( 1 + (-0.701 + 0.712i)T \)
19 \( 1 + (0.277 + 0.960i)T \)
23 \( 1 + (-0.340 - 0.940i)T \)
29 \( 1 + (0.945 + 0.324i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (-0.956 - 0.293i)T \)
41 \( 1 + (0.879 - 0.475i)T \)
43 \( 1 + (0.997 - 0.0660i)T \)
47 \( 1 + (0.986 - 0.164i)T \)
53 \( 1 + (-0.980 - 0.197i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (0.490 + 0.871i)T \)
67 \( 1 + (-0.980 - 0.197i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.956 + 0.293i)T \)
79 \( 1 + (-0.922 - 0.386i)T \)
83 \( 1 + (-0.148 + 0.988i)T \)
89 \( 1 + (0.148 - 0.988i)T \)
97 \( 1 + (-0.518 + 0.854i)T \)
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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

−22.8607613377541316007665028283, −22.00027107230577492348858277219, −21.778618341474171054627683138463, −20.66384493578204032428621520057, −19.39031176999026380428482049567, −18.24705209838840024903803735778, −17.735573205132164806583305875712, −17.03236223901326807458486599884, −15.99638812184203201383767368160, −15.56431442415380654840034523538, −14.13855364154823139481585833107, −13.46251006769089882135082469785, −12.742692494735028068813165487622, −12.109465462997214515583227393937, −11.0921762570564887405255802506, −9.71116781246475475809836554783, −9.02258461145127495757263260577, −7.54394741934114232965675244503, −6.83036731018124643443979564857, −6.08096799753276669539939878263, −5.22762980691238042933780864533, −4.60910562146485068666518110427, −2.76171336680670528564144496233, −2.047700739247173177966306564035, −0.05848019450788368055778589471, 1.02830207533440860023780127145, 2.499113747219319072875461736438, 3.47704679159336407831565513601, 4.483798889561131543586789370394, 5.60987049718187854853632510569, 5.98803434911795193479548332059, 7.05641983811441048378335216704, 8.87804686352271962594356477764, 9.946127893751253407718224451642, 10.505873748518433628715807440083, 10.83775717101997510529154571509, 12.43449486669279736132104752521, 12.747895009875283759275963453738, 13.79393686326820340890600315858, 14.57501655215036204989264264031, 15.68925647467532517363070103128, 16.411452488808754338510418935591, 17.45878635988563194996741062106, 18.16622713078122788498167548842, 19.13776806135013095817224784155, 20.23519107022261579352279721048, 20.88434090499102661815449708368, 21.69421345907513084500091349855, 22.36611129540986633450137922685, 22.83357370301110325558690202472

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