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Properties

Label 1-571-571.254-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.874 + 0.484i$
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 + (0.956 − 0.293i)3-s + (−0.677 + 0.735i)4-s + (0.997 + 0.0660i)5-s + (0.652 + 0.757i)6-s + (0.934 + 0.355i)7-s + (−0.945 − 0.324i)8-s + (0.828 − 0.560i)9-s + (0.340 + 0.940i)10-s + (−0.340 + 0.940i)11-s + (−0.431 + 0.901i)12-s + (−0.461 + 0.887i)13-s + (0.0495 + 0.998i)14-s + (0.973 − 0.229i)15-s + (−0.0825 − 0.996i)16-s + (−0.922 + 0.386i)17-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)2-s + (0.956 − 0.293i)3-s + (−0.677 + 0.735i)4-s + (0.997 + 0.0660i)5-s + (0.652 + 0.757i)6-s + (0.934 + 0.355i)7-s + (−0.945 − 0.324i)8-s + (0.828 − 0.560i)9-s + (0.340 + 0.940i)10-s + (−0.340 + 0.940i)11-s + (−0.431 + 0.901i)12-s + (−0.461 + 0.887i)13-s + (0.0495 + 0.998i)14-s + (0.973 − 0.229i)15-s + (−0.0825 − 0.996i)16-s + (−0.922 + 0.386i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.874 + 0.484i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.874 + 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9396561248 + 3.633442873i\)
\(L(\frac12)\) \(\approx\) \(0.9396561248 + 3.633442873i\)
\(L(1)\) \(\approx\) \(1.500551176 + 1.232038682i\)
\(L(1)\) \(\approx\) \(1.500551176 + 1.232038682i\)

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 \( 1 + (0.956 - 0.293i)T \)
5 \( 1 + (0.997 + 0.0660i)T \)
7 \( 1 + (0.934 + 0.355i)T \)
11 \( 1 + (-0.340 + 0.940i)T \)
13 \( 1 + (-0.461 + 0.887i)T \)
17 \( 1 + (-0.922 + 0.386i)T \)
19 \( 1 + (-0.601 + 0.799i)T \)
23 \( 1 + (-0.574 + 0.818i)T \)
29 \( 1 + (-0.986 - 0.164i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (-0.148 + 0.988i)T \)
41 \( 1 + (-0.245 - 0.969i)T \)
43 \( 1 + (-0.999 + 0.0330i)T \)
47 \( 1 + (0.0825 + 0.996i)T \)
53 \( 1 + (0.995 + 0.0990i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (0.863 + 0.504i)T \)
67 \( 1 + (0.995 + 0.0990i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.148 - 0.988i)T \)
79 \( 1 + (-0.980 - 0.197i)T \)
83 \( 1 + (0.652 + 0.757i)T \)
89 \( 1 + (-0.652 - 0.757i)T \)
97 \( 1 + (0.490 + 0.871i)T \)
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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

−22.19881685610961629242164244133, −21.668395424627364966196319925970, −21.079275135092865650147121458727, −20.20210877537887604385941472470, −19.830208766920765658761815844573, −18.5139129603950222860112778787, −18.0182793259243022464363011538, −16.99022944472685196599209603946, −15.65026146769724297487663356629, −14.65148260095598657094353170906, −14.161224575943804105390766172755, −13.287040207370054201980386318848, −12.85854384803104773673621192251, −11.303126831443743557756288962522, −10.59162584940455541955882267030, −9.92910450192566941171157304652, −8.81725416822312620794287385173, −8.33416561188645401996115469580, −6.83131995419612198405156109273, −5.38606237167646390567311434114, −4.80471387327317609746726903506, −3.65460798890959290081750646300, −2.54316319465668559872972685654, −1.97288141400207580572313911906, −0.626486053911535416831841484201, 1.81955050730628185078406020736, 2.30559506454774395841484917061, 3.90843536346735146507245769552, 4.73380710916377390476991316836, 5.80856436768178765758025910019, 6.80837483490099228036482792313, 7.61728866264332688697652466723, 8.514862924896579869640540555, 9.297005562071002998926850965720, 10.08971831308597555257652290215, 11.72722010079654589460890739939, 12.70968540678780393178955143356, 13.45092290223582849508676470247, 14.17419808663002452304757317122, 14.92504783659518911961628421064, 15.35345275766356804175314536038, 16.78629805208138888824168503538, 17.53421151709346059988946161004, 18.21841745436314585431255967694, 18.94467158478384566150563126074, 20.37523787317478495767641812505, 21.03636531955681593343805649648, 21.67589018556906717776406111992, 22.50162766400558993035467577910, 23.94109148126489138703173676240

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