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Properties

Label 1-571-571.177-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.422 - 0.906i$
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  + (−0.451 + 0.892i)2-s + (0.782 + 0.622i)3-s + (−0.592 − 0.805i)4-s + (0.528 + 0.849i)5-s + (−0.909 + 0.416i)6-s + (−0.180 − 0.983i)7-s + (0.986 − 0.164i)8-s + (0.224 + 0.974i)9-s + (−0.996 + 0.0880i)10-s + (−0.421 − 0.906i)11-s + (0.0385 − 0.999i)12-s + (−0.480 − 0.876i)13-s + (0.959 + 0.282i)14-s + (−0.115 + 0.993i)15-s + (−0.298 + 0.954i)16-s + (0.319 + 0.947i)17-s + ⋯
L(s)  = 1  + (−0.451 + 0.892i)2-s + (0.782 + 0.622i)3-s + (−0.592 − 0.805i)4-s + (0.528 + 0.849i)5-s + (−0.909 + 0.416i)6-s + (−0.180 − 0.983i)7-s + (0.986 − 0.164i)8-s + (0.224 + 0.974i)9-s + (−0.996 + 0.0880i)10-s + (−0.421 − 0.906i)11-s + (0.0385 − 0.999i)12-s + (−0.480 − 0.876i)13-s + (0.959 + 0.282i)14-s + (−0.115 + 0.993i)15-s + (−0.298 + 0.954i)16-s + (0.319 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2117546010 + 0.3322301538i\)
\(L(\frac12)\) \(\approx\) \(-0.2117546010 + 0.3322301538i\)
\(L(1)\) \(\approx\) \(0.7280052546 + 0.5179644365i\)
\(L(1)\) \(\approx\) \(0.7280052546 + 0.5179644365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.451 + 0.892i)T \)
3 \( 1 + (0.782 + 0.622i)T \)
5 \( 1 + (0.528 + 0.849i)T \)
7 \( 1 + (-0.180 - 0.983i)T \)
11 \( 1 + (-0.421 - 0.906i)T \)
13 \( 1 + (-0.480 - 0.876i)T \)
17 \( 1 + (0.319 + 0.947i)T \)
19 \( 1 + (0.834 - 0.551i)T \)
23 \( 1 + (-0.461 + 0.887i)T \)
29 \( 1 + (-0.821 + 0.569i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.982 - 0.186i)T \)
41 \( 1 + (0.926 + 0.376i)T \)
43 \( 1 + (-0.857 - 0.514i)T \)
47 \( 1 + (-0.975 + 0.218i)T \)
53 \( 1 + (0.889 + 0.456i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (-0.256 + 0.966i)T \)
67 \( 1 + (-0.840 + 0.542i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.329 - 0.944i)T \)
79 \( 1 + (-0.411 + 0.911i)T \)
83 \( 1 + (0.0935 - 0.995i)T \)
89 \( 1 + (-0.815 - 0.578i)T \)
97 \( 1 + (0.00551 + 0.999i)T \)
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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

−22.25301533028943329093388996223, −21.25225812781038013959526112710, −20.70762178028390356224300149232, −20.06432241644718879269070011986, −19.16621342173218988854273135155, −18.293932971257256079212712341123, −17.96175067158754411870253381489, −16.74375075537122817247329419981, −15.89411633472671963304804574822, −14.53985774097527091123783719521, −13.79730056034800649682800795118, −12.81320066095802828386210311184, −12.29608924054894579247404762749, −11.69396792622697246086990147088, −9.920062940016537837679912352997, −9.50881280027394419766899650777, −8.76945894903201340778704921492, −7.90133493122876376743964794005, −6.92698338338740704214802792664, −5.43137906900406883757137064422, −4.44620281638660548474103523857, −3.104290442152891602147401111572, −2.137014102540335765281406386389, −1.6031060882419039055970655308, −0.09347285165379957582966561878, 1.538837744900047239175068119559, 3.04785091906407088590875812826, 3.83682829288325914197321636588, 5.232456824883918980359899702648, 5.97645243723829500522760781291, 7.39396958971663122229086762879, 7.66571777319916880519342148264, 8.8873693502620866307373851779, 9.82242150823001569696874052114, 10.40789384269070733785747173140, 11.03206694519399962796156566795, 13.26948680385096864523259901669, 13.575913763907360042405717635988, 14.5359657722913579402838039265, 15.09101440120684667110739925463, 16.03277671819555573043987899445, 16.80550331556340003047551344081, 17.65816290405969239969105711655, 18.536178667509419103863422183169, 19.47709791263213527488695962904, 19.98975735189076709559402258545, 21.19820202012816939317602603385, 22.09985717284965385844012165160, 22.68534644961927168233064961383, 23.83513106746423625498343940875

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