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Properties

Label 1-539-539.25-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.186 + 0.982i$
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  + (−0.280 − 0.959i)2-s + (0.193 − 0.981i)3-s + (−0.842 + 0.538i)4-s + (0.0149 + 0.999i)5-s + (−0.995 + 0.0896i)6-s + (0.753 + 0.657i)8-s + (−0.925 − 0.379i)9-s + (0.955 − 0.294i)10-s + (0.365 + 0.930i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (0.420 − 0.907i)16-s + (−0.163 − 0.986i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (−0.550 − 0.834i)20-s + ⋯
L(s)  = 1  + (−0.280 − 0.959i)2-s + (0.193 − 0.981i)3-s + (−0.842 + 0.538i)4-s + (0.0149 + 0.999i)5-s + (−0.995 + 0.0896i)6-s + (0.753 + 0.657i)8-s + (−0.925 − 0.379i)9-s + (0.955 − 0.294i)10-s + (0.365 + 0.930i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (0.420 − 0.907i)16-s + (−0.163 − 0.986i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (−0.550 − 0.834i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.186 + 0.982i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.186 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01802324928 + 0.02177100581i\)
\(L(\frac12)\) \(\approx\) \(0.01802324928 + 0.02177100581i\)
\(L(1)\) \(\approx\) \(0.5361210254 - 0.3386008971i\)
\(L(1)\) \(\approx\) \(0.5361210254 - 0.3386008971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.280 - 0.959i)T \)
3 \( 1 + (0.193 - 0.981i)T \)
5 \( 1 + (0.0149 + 0.999i)T \)
13 \( 1 + (-0.691 + 0.722i)T \)
17 \( 1 + (-0.163 - 0.986i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.251 - 0.967i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.337 + 0.941i)T \)
53 \( 1 + (-0.772 - 0.635i)T \)
59 \( 1 + (-0.946 - 0.323i)T \)
61 \( 1 + (-0.772 + 0.635i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (-0.337 - 0.941i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.691 - 0.722i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
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

−24.167250316116789425110819037900, −23.327907190652344071078508887820, −22.313265130168779232478722671394, −21.734263381536409609935522908354, −20.53508643090852139271849815901, −19.97494527644853699601296897254, −19.00594789924587935700606269462, −17.781767064620343252770180077315, −16.89432914961166080642310596461, −16.582648169592770734653853940721, −15.539622337154624223043164081648, −14.97620343537392174704274601598, −14.080309459358554813094394951656, −13.07963185552300797712201045920, −12.15334736105749143098410311308, −10.65266200948515232958234947832, −9.92148802916218294016850679681, −9.14674300031609479857793132478, −8.2742189753371761357760966383, −7.679363932779556667971298384113, −5.98720756973107368134600013979, −5.44404856614999733185856813786, −4.43148661111498899268486542485, −3.69244693271238878708592076488, −1.81830821655197875171098978884, 0.0151376998434528631618447234, 1.72589787513676750200679611096, 2.51555645610167496965054463071, 3.32455764162783865748986566054, 4.63834092257644740791221297298, 6.05167137525960354827277122820, 7.2309598370279766603810261296, 7.68311362426832698883696849947, 9.06589819003066171396002512697, 9.66049236315009674728946660599, 11.051028292006532044411755837, 11.44587768280520335621187013148, 12.41592726592547337412689899824, 13.29888571709227615281097245834, 14.14228627727731524134242499745, 14.68197653112108459171644437486, 16.243664274870239646200239813757, 17.40286051006933195921246205260, 18.10621961643764965825094109244, 18.57691124925105663986650681538, 19.59473706371055175339743396283, 19.87865550993414273906647867179, 21.12341363909809464548231264309, 22.03306089238169593275479151981, 22.64294733899266042082212227730

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