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Properties

Label 1-539-539.235-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.0899 + 0.995i$
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  + (0.0149 + 0.999i)2-s + (−0.842 + 0.538i)3-s + (−0.999 + 0.0299i)4-s + (−0.163 + 0.986i)5-s + (−0.550 − 0.834i)6-s + (−0.0448 − 0.998i)8-s + (0.420 − 0.907i)9-s + (−0.988 − 0.149i)10-s + (0.826 − 0.563i)12-s + (0.858 − 0.512i)13-s + (−0.393 − 0.919i)15-s + (0.998 − 0.0598i)16-s + (0.971 + 0.237i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (0.134 − 0.990i)20-s + ⋯
L(s)  = 1  + (0.0149 + 0.999i)2-s + (−0.842 + 0.538i)3-s + (−0.999 + 0.0299i)4-s + (−0.163 + 0.986i)5-s + (−0.550 − 0.834i)6-s + (−0.0448 − 0.998i)8-s + (0.420 − 0.907i)9-s + (−0.988 − 0.149i)10-s + (0.826 − 0.563i)12-s + (0.858 − 0.512i)13-s + (−0.393 − 0.919i)15-s + (0.998 − 0.0598i)16-s + (0.971 + 0.237i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (0.134 − 0.990i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.0899 + 0.995i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.0899 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6437884675 + 0.7045203732i\)
\(L(\frac12)\) \(\approx\) \(0.6437884675 + 0.7045203732i\)
\(L(1)\) \(\approx\) \(0.6379965731 + 0.5105631139i\)
\(L(1)\) \(\approx\) \(0.6379965731 + 0.5105631139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0149 + 0.999i)T \)
3 \( 1 + (-0.842 + 0.538i)T \)
5 \( 1 + (-0.163 + 0.986i)T \)
13 \( 1 + (0.858 - 0.512i)T \)
17 \( 1 + (0.971 + 0.237i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.337 + 0.941i)T \)
41 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.599 - 0.800i)T \)
53 \( 1 + (-0.280 + 0.959i)T \)
59 \( 1 + (0.887 - 0.460i)T \)
61 \( 1 + (-0.280 - 0.959i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (-0.599 + 0.800i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.858 + 0.512i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
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

−23.262769780804649285175817621840, −22.429366954146239998306741263246, −21.200429991651671790636591839911, −20.99707399062604714498251997583, −19.65533339077457132723365611031, −19.21117402123108350326002643908, −18.085414722534085977106435046160, −17.6284200591725021181664504315, −16.476354372977439677365325161822, −15.99200832837927381895828917253, −14.23600739161245323313666373234, −13.477085239674600301241381695900, −12.71147849795529409276165362415, −11.88209334391063538438263934581, −11.47352642115829750559908702889, −10.30777039901878399928550653684, −9.434000930827261075238026548181, −8.38439814144707409466784850268, −7.52919400429766593410489777648, −6.00435911484439700617471807236, −5.21487171501527725879448078874, −4.358685324915850407642061942606, −3.17516083511736151201841243618, −1.543382125561048534386349521149, −1.03034467829344266023015802569, 0.81420785660119148407019956370, 3.12266874151741144851261745069, 3.98147733439156391376238578311, 5.10349102869216839043408397939, 6.00277745928400878274944674044, 6.658366913458665589728354898931, 7.63443426958057453015479119939, 8.66947912318552200420852964320, 9.94675468760151703911458289789, 10.41840598645034196254149783088, 11.53919473923371849458725181357, 12.471310856440570373392903039097, 13.682505884905186425833971748535, 14.479108599828672879555472555951, 15.47461732996289061213433128637, 15.80466975276334900344372026479, 16.87749105714869877326160591424, 17.60933661644850280869234677682, 18.417701896703618338247052469532, 18.951744476078610173455178558145, 20.502272585225973225337344681461, 21.50013109972564788413648645521, 22.3306640618780456027949937050, 22.84510047392169266405056638828, 23.45824784385928037933337368543

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