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Properties

Label 1-539-539.13-r0-0-0
Degree $1$
Conductor $539$
Sign $0.996 + 0.0805i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (−0.983 − 0.178i)2-s + (−0.858 − 0.512i)3-s + (0.936 + 0.351i)4-s + (0.393 + 0.919i)5-s + (0.753 + 0.657i)6-s + (−0.858 − 0.512i)8-s + (0.473 + 0.880i)9-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (0.753 + 0.657i)16-s + (−0.963 − 0.266i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.0448 + 0.998i)20-s + ⋯
L(s)  = 1  + (−0.983 − 0.178i)2-s + (−0.858 − 0.512i)3-s + (0.936 + 0.351i)4-s + (0.393 + 0.919i)5-s + (0.753 + 0.657i)6-s + (−0.858 − 0.512i)8-s + (0.473 + 0.880i)9-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (0.753 + 0.657i)16-s + (−0.963 − 0.266i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.0448 + 0.998i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.996 + 0.0805i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.996 + 0.0805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7155348758 + 0.02885057173i\)
\(L(\frac12)\) \(\approx\) \(0.7155348758 + 0.02885057173i\)
\(L(1)\) \(\approx\) \(0.6273103223 + 0.02317262281i\)
\(L(1)\) \(\approx\) \(0.6273103223 + 0.02317262281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.983 - 0.178i)T \)
3 \( 1 + (-0.858 - 0.512i)T \)
5 \( 1 + (0.393 + 0.919i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (-0.963 - 0.266i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.550 - 0.834i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.550 + 0.834i)T \)
41 \( 1 + (0.858 + 0.512i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.134 - 0.990i)T \)
53 \( 1 + (-0.963 + 0.266i)T \)
59 \( 1 + (-0.858 + 0.512i)T \)
61 \( 1 + (-0.963 - 0.266i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (0.134 - 0.990i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−23.58968908818814036348549561862, −22.66472111707732070553455132744, −21.44159218266782079779297970918, −20.87295211235958170580294018778, −20.18357599392186496185567567399, −19.07469129617149212800383598779, −18.03584016463852103982566812044, −17.4700790092729262384103126266, −16.7933134160882035705192644506, −15.86043669265815683339471539295, −15.57686449009144828192823388820, −14.14470603252319364045313553600, −12.87091762359734692273652851169, −12.04153221853499408260962942317, −11.061837242271943824220396900912, −10.40942696747775470289452001629, −9.36829630436516363396524669513, −8.83114437354615043767475686298, −7.71428993955573038536640403343, −6.41180291971293044402121822941, −5.80535278488310459706155726197, −4.83339432093007651880548874515, −3.54550607693722872718409034184, −1.79742576590434774873259985232, −0.817873853418124067045879339687, 0.92631743743434678798087281646, 2.13039073411511221837770320267, 3.05698507934835960220225524771, 4.71202133039525227777839575582, 6.28931416418399716564544741460, 6.51065741354996001500524410609, 7.48756069883649128368338175787, 8.56537970578107419740853164650, 9.644237805871270711002404805516, 10.67115840231720588738335919656, 11.11669978399178652563402799121, 11.887223009874190909347394541326, 13.0778313203191322273435525763, 13.85994713627313293006089830953, 15.31725785013513989812162938277, 15.94890680722711473419644069566, 17.017213017332755214398254388480, 17.74575697911029092893041494714, 18.25374355437443454184907046398, 18.98034981273884141840663309343, 19.764825806217988513726697536672, 20.985185793177017686517489595546, 21.72405388254697761445968304510, 22.60498164251887793587900336934, 23.38083978654159117522010112378

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