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Properties

Label 1-539-539.102-r0-0-0
Degree $1$
Conductor $539$
Sign $0.874 - 0.485i$
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  + (0.998 − 0.0598i)2-s + (−0.646 − 0.762i)3-s + (0.992 − 0.119i)4-s + (0.791 + 0.611i)5-s + (−0.691 − 0.722i)6-s + (0.983 − 0.178i)8-s + (−0.163 + 0.986i)9-s + (0.826 + 0.563i)10-s + (−0.733 − 0.680i)12-s + (−0.550 − 0.834i)13-s + (−0.0448 − 0.998i)15-s + (0.971 − 0.237i)16-s + (0.575 + 0.817i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.858 + 0.512i)20-s + ⋯
L(s)  = 1  + (0.998 − 0.0598i)2-s + (−0.646 − 0.762i)3-s + (0.992 − 0.119i)4-s + (0.791 + 0.611i)5-s + (−0.691 − 0.722i)6-s + (0.983 − 0.178i)8-s + (−0.163 + 0.986i)9-s + (0.826 + 0.563i)10-s + (−0.733 − 0.680i)12-s + (−0.550 − 0.834i)13-s + (−0.0448 − 0.998i)15-s + (0.971 − 0.237i)16-s + (0.575 + 0.817i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.858 + 0.512i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.427721038 - 0.6286564094i\)
\(L(\frac12)\) \(\approx\) \(2.427721038 - 0.6286564094i\)
\(L(1)\) \(\approx\) \(1.810750805 - 0.3272208313i\)
\(L(1)\) \(\approx\) \(1.810750805 - 0.3272208313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.998 - 0.0598i)T \)
3 \( 1 + (-0.646 - 0.762i)T \)
5 \( 1 + (0.791 + 0.611i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (0.575 + 0.817i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.193 + 0.981i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.842 - 0.538i)T \)
53 \( 1 + (0.420 + 0.907i)T \)
59 \( 1 + (-0.337 - 0.941i)T \)
61 \( 1 + (0.420 - 0.907i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.842 + 0.538i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.550 + 0.834i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−23.32758544263416455135092107810, −22.62539182041613422581499206504, −21.75516215570484152368972847068, −21.14178622652012802999470015346, −20.67442604782232394820803849287, −19.62780631094712738845760333677, −18.27701283359457704272996263201, −17.21510064829380132188269341408, −16.38002877479946030546275354184, −16.20877121152412033530475643513, −14.68028871882751332866416315639, −14.36925129800070332946771700414, −13.1393319720547053752677723675, −12.32404465979686380448658298428, −11.632128719656941145966799892230, −10.580040008630116430438946055074, −9.77194359208062718189612163959, −8.840351987766614063171529591767, −7.29013258482280670903892458923, −6.31277034664248192346526107117, −5.42989766305299901615910545424, −4.83901457237263567945307938161, −3.903495681777575802825887220955, −2.672237221731881050805901063821, −1.34291669665923560333473373223, 1.28557427904346573214850151106, 2.38245961865167334377772352639, 3.2122637247359704888846426372, 4.84326777739085847668004925722, 5.54888512204275397973163715757, 6.40569148172865535128448460374, 7.09799218816916563223566336288, 8.05722667269538808129368161711, 9.80173981845163962186877866514, 10.70630724569497899762865952804, 11.32021430756733103746094927567, 12.450024367879616232127297060339, 13.04876345947098230048202191230, 13.773074158735815577764073346484, 14.71387993963432145593913308277, 15.45758926125064305039245199114, 16.80713251976713781335586722742, 17.32317783120934106295846781678, 18.2740511378488254346191010873, 19.27122716994970192557861490635, 19.94962909344155093603907252505, 21.28035607159519281842534403637, 21.80354087207287787212225390384, 22.58824741441928649820467365669, 23.2500456370384026361510793159

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