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Properties

Label 1-539-539.141-r0-0-0
Degree $1$
Conductor $539$
Sign $0.953 - 0.302i$
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  + (−0.393 + 0.919i)2-s + (0.936 − 0.351i)3-s + (−0.691 − 0.722i)4-s + (−0.963 − 0.266i)5-s + (−0.0448 + 0.998i)6-s + (0.936 − 0.351i)8-s + (0.753 − 0.657i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)12-s + (−0.393 + 0.919i)13-s + (−0.995 + 0.0896i)15-s + (−0.0448 + 0.998i)16-s + (0.983 − 0.178i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (0.473 + 0.880i)20-s + ⋯
L(s)  = 1  + (−0.393 + 0.919i)2-s + (0.936 − 0.351i)3-s + (−0.691 − 0.722i)4-s + (−0.963 − 0.266i)5-s + (−0.0448 + 0.998i)6-s + (0.936 − 0.351i)8-s + (0.753 − 0.657i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)12-s + (−0.393 + 0.919i)13-s + (−0.995 + 0.0896i)15-s + (−0.0448 + 0.998i)16-s + (0.983 − 0.178i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (0.473 + 0.880i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.953 - 0.302i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.953 - 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143071278 - 0.1767754689i\)
\(L(\frac12)\) \(\approx\) \(1.143071278 - 0.1767754689i\)
\(L(1)\) \(\approx\) \(0.9707734109 + 0.09682255295i\)
\(L(1)\) \(\approx\) \(0.9707734109 + 0.09682255295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.393 + 0.919i)T \)
3 \( 1 + (0.936 - 0.351i)T \)
5 \( 1 + (-0.963 - 0.266i)T \)
13 \( 1 + (-0.393 + 0.919i)T \)
17 \( 1 + (0.983 - 0.178i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.134 - 0.990i)T \)
41 \( 1 + (0.936 - 0.351i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.995 - 0.0896i)T \)
53 \( 1 + (0.983 + 0.178i)T \)
59 \( 1 + (0.936 + 0.351i)T \)
61 \( 1 + (0.983 - 0.178i)T \)
67 \( 1 + T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (-0.995 + 0.0896i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.393 - 0.919i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−23.20404300746512541764412795317, −22.434485021144873461770020728223, −21.642221541116671353205521270538, −20.66851677937520429894060852643, −20.07960297960518063820155827501, −19.46783860106323955087465103019, −18.665610542084117479741690011188, −17.976210922802743787143199082873, −16.5393172922443117860933661229, −15.97894679621559412205070650280, −14.6527404674210118363889460876, −14.30268258370782127200763223284, −12.88019406513527630396363417546, −12.364586142303809880802211550346, −11.2708460194961643798336200161, −10.28029433025713162294753590200, −9.80193666188813735872959465440, −8.46396147430503981700963324693, −8.02886045218029541404961379394, −7.20935919901108617903438523107, −5.26124540617279538602586135949, −4.10749125827560213041040374673, −3.39877461059447166437029716577, −2.648879763339487736851774554227, −1.255974636584169936796668369676, 0.724340882299287042935233856968, 2.15001428892689382815826117398, 3.705212324368019084548092904560, 4.432516074968896819769913375605, 5.71267779277345996025424181421, 7.080614290757310468674212994016, 7.486587492709423936816843660703, 8.35859769859952926016981750815, 9.19906655721923849951313022181, 9.88902042368839936905943429727, 11.35992020692252042803968706343, 12.36062200075970553666195703063, 13.37885567151527462294457019539, 14.24592991646612603687261456237, 14.888817092812907168939737129739, 15.8018592954963560188243598570, 16.36346515858247864779075256046, 17.497118012643402869822203811176, 18.451973264759289356912122347635, 19.25138329886284502892029487180, 19.64198453714481378655817823632, 20.640821269567936786706602182543, 21.75014793555849908445649838827, 22.918079117364901045869467861226, 23.76735631635510955476668979914

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