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Properties

Label 1-571-571.86-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.763 - 0.645i$
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  + (0.401 + 0.915i)2-s + (0.0165 − 0.999i)3-s + (−0.677 + 0.735i)4-s + (−0.768 − 0.639i)5-s + (0.922 − 0.386i)6-s + (0.627 − 0.778i)7-s + (−0.945 − 0.324i)8-s + (−0.999 − 0.0330i)9-s + (0.277 − 0.960i)10-s + (−0.277 − 0.960i)11-s + (0.724 + 0.689i)12-s + (0.894 − 0.446i)13-s + (0.965 + 0.261i)14-s + (−0.652 + 0.757i)15-s + (−0.0825 − 0.996i)16-s + (0.518 − 0.854i)17-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)2-s + (0.0165 − 0.999i)3-s + (−0.677 + 0.735i)4-s + (−0.768 − 0.639i)5-s + (0.922 − 0.386i)6-s + (0.627 − 0.778i)7-s + (−0.945 − 0.324i)8-s + (−0.999 − 0.0330i)9-s + (0.277 − 0.960i)10-s + (−0.277 − 0.960i)11-s + (0.724 + 0.689i)12-s + (0.894 − 0.446i)13-s + (0.965 + 0.261i)14-s + (−0.652 + 0.757i)15-s + (−0.0825 − 0.996i)16-s + (0.518 − 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5175387152 - 1.414258172i\)
\(L(\frac12)\) \(\approx\) \(0.5175387152 - 1.414258172i\)
\(L(1)\) \(\approx\) \(1.038888602 - 0.2711606625i\)
\(L(1)\) \(\approx\) \(1.038888602 - 0.2711606625i\)

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 \( 1 + (0.0165 - 0.999i)T \)
5 \( 1 + (-0.768 - 0.639i)T \)
7 \( 1 + (0.627 - 0.778i)T \)
11 \( 1 + (-0.277 - 0.960i)T \)
13 \( 1 + (0.894 - 0.446i)T \)
17 \( 1 + (0.518 - 0.854i)T \)
19 \( 1 + (0.956 - 0.293i)T \)
23 \( 1 + (0.601 + 0.799i)T \)
29 \( 1 + (-0.986 - 0.164i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (0.701 - 0.712i)T \)
41 \( 1 + (-0.245 - 0.969i)T \)
43 \( 1 + (-0.340 - 0.940i)T \)
47 \( 1 + (0.0825 + 0.996i)T \)
53 \( 1 + (-0.863 + 0.504i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (0.746 - 0.665i)T \)
67 \( 1 + (-0.863 + 0.504i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.701 + 0.712i)T \)
79 \( 1 + (-0.490 + 0.871i)T \)
83 \( 1 + (0.922 - 0.386i)T \)
89 \( 1 + (-0.922 + 0.386i)T \)
97 \( 1 + (0.115 - 0.993i)T \)
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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

−23.19182317145174969755059283517, −22.32899373508999074669493958163, −21.72933744803369187584708291884, −20.81759432891253562846036265332, −20.36618905836219172204068071959, −19.32590951707034763257842871767, −18.4628891991541342904359171705, −17.86752837780984218876493022367, −16.44519329369786158003327191669, −15.46814340885175787733800463573, −14.7603091260429374841574190713, −14.42468978810803736837282378020, −13.00419337333349574577657431101, −11.926131430092245802181768416114, −11.4034249479774525623823300366, −10.61145905303765157969945019376, −9.82455249790404506872480787548, −8.79906798096650034369703693662, −7.99526906616949527450208327323, −6.373982643425498995410422093312, −5.27027058789886517345416566742, −4.50128147311371425735801827303, −3.57620875211425652338656396488, −2.79113869954051558215746704089, −1.528548601198828237934519794690, 0.38021753009661700907944456607, 1.0959377545231022613276078955, 3.09206827581299476006120086231, 3.911681819925773731780009787783, 5.24232242082036277643561885277, 5.7914488810987443781250508257, 7.28104330090286065027679097549, 7.59803077634919206217650473471, 8.39613407171835121233464440606, 9.24670598541179382615898832457, 11.17230317172390551118174407047, 11.63807434731863172725310762467, 12.80603704630720935324980342820, 13.50729520262890629663517208546, 13.99283287502059449471502265468, 15.13297513452108185433198723859, 16.08636897294317153212672380033, 16.751251094847632796995719169569, 17.57948189820247239401197211343, 18.4386988811858866112886576036, 19.16536044527848334797648736080, 20.482406070950597534478757737803, 20.78121750503062361204087996045, 22.28927622011203067919816211108, 23.14257444370236135640990725927

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