Properties

Label 1-571-571.2-r1-0-0
Degree $1$
Conductor $571$
Sign $0.444 + 0.895i$
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  + (0.821 + 0.569i)2-s + (0.962 − 0.272i)3-s + (0.350 + 0.936i)4-s + (−0.998 + 0.0550i)5-s + (0.945 + 0.324i)6-s + (0.677 − 0.735i)7-s + (−0.245 + 0.969i)8-s + (0.851 − 0.523i)9-s + (−0.851 − 0.523i)10-s + (0.0275 + 0.999i)11-s + (0.592 + 0.805i)12-s + (−0.926 − 0.376i)13-s + (0.975 − 0.218i)14-s + (−0.945 + 0.324i)15-s + (−0.754 + 0.656i)16-s + (0.754 − 0.656i)17-s + ⋯
L(s)  = 1  + (0.821 + 0.569i)2-s + (0.962 − 0.272i)3-s + (0.350 + 0.936i)4-s + (−0.998 + 0.0550i)5-s + (0.945 + 0.324i)6-s + (0.677 − 0.735i)7-s + (−0.245 + 0.969i)8-s + (0.851 − 0.523i)9-s + (−0.851 − 0.523i)10-s + (0.0275 + 0.999i)11-s + (0.592 + 0.805i)12-s + (−0.926 − 0.376i)13-s + (0.975 − 0.218i)14-s + (−0.945 + 0.324i)15-s + (−0.754 + 0.656i)16-s + (0.754 − 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.952891522 + 2.452336442i\)
\(L(\frac12)\) \(\approx\) \(3.952891522 + 2.452336442i\)
\(L(1)\) \(\approx\) \(2.117238959 + 0.7260744026i\)
\(L(1)\) \(\approx\) \(2.117238959 + 0.7260744026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.821 + 0.569i)T \)
3 \( 1 + (0.962 - 0.272i)T \)
5 \( 1 + (-0.998 + 0.0550i)T \)
7 \( 1 + (0.677 - 0.735i)T \)
11 \( 1 + (0.0275 + 0.999i)T \)
13 \( 1 + (-0.926 - 0.376i)T \)
17 \( 1 + (0.754 - 0.656i)T \)
19 \( 1 + (0.962 - 0.272i)T \)
23 \( 1 + (0.245 + 0.969i)T \)
29 \( 1 + (0.137 + 0.990i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (-0.926 + 0.376i)T \)
41 \( 1 + (0.998 - 0.0550i)T \)
43 \( 1 + (0.851 + 0.523i)T \)
47 \( 1 + (0.191 + 0.981i)T \)
53 \( 1 + (0.821 - 0.569i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (0.904 - 0.426i)T \)
67 \( 1 + (-0.904 - 0.426i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.137 + 0.990i)T \)
79 \( 1 + (-0.635 - 0.771i)T \)
83 \( 1 + (-0.191 - 0.981i)T \)
89 \( 1 + (0.754 - 0.656i)T \)
97 \( 1 + (0.350 + 0.936i)T \)
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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

−22.68942619554376933014265431607, −21.94459950112676944900810740204, −21.08314701657713194419206715529, −20.66518590012748997875674579037, −19.444381831090865509631598311175, −19.194188943137383814109781965849, −18.4212251850496427090945955482, −16.67698233053026335922516811866, −15.759389043621610828925108484094, −15.097429790798846783417696345550, −14.39596035636200165682213892708, −13.77873197719541450808295258402, −12.46929715006305454618414354656, −11.9695250755112713652486143575, −11.026345640747530685145371886142, −10.07775820012781511916218325445, −8.98552911586830064687731777628, −8.1502338703878387839444472692, −7.2546845669546692348061300022, −5.80444606330910624051359368658, −4.784559939253035244796833593274, −3.98313780132698923726156996321, −3.0593708188840814909548357869, −2.234108598252786345004841939120, −0.87048710266960570458410044087, 1.16653263806709977303416864991, 2.67751172508290879779946233433, 3.47144889461002020652983352353, 4.47418757252204096465208938488, 5.11716138688100850118720879378, 6.97437796385607727016058160353, 7.47251203121014977217325124718, 7.84692715092045415357061883460, 9.07301985718709388412566106765, 10.25626232597205070416785688541, 11.61473958713733262217238191446, 12.23912782484521949277252553165, 13.096242912580295395419048795489, 14.21452006796282601346721541063, 14.52357648716071880502912884801, 15.45936415450264335624739649621, 16.059856400987334278929771861699, 17.33671628110297085740435708014, 17.99043078770658964247397647333, 19.31806854090588078791879183873, 20.1178268270115453633953593723, 20.55215290004559455753888511115, 21.458050717619554730364655669102, 22.72535625949928650392084862857, 23.17886473804875655066789973711

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