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Properties

Label 1-547-547.409-r0-0-0
Degree $1$
Conductor $547$
Sign $0.308 + 0.951i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.449 − 0.893i)2-s + (−0.900 − 0.433i)3-s + (−0.596 − 0.802i)4-s + (−0.985 − 0.171i)5-s + (−0.792 + 0.609i)6-s + (0.770 − 0.636i)7-s + (−0.985 + 0.171i)8-s + (0.623 + 0.781i)9-s + (−0.596 + 0.802i)10-s + (−0.354 − 0.935i)11-s + (0.188 + 0.982i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.813 + 0.582i)15-s + (−0.289 + 0.957i)16-s + (−0.952 + 0.305i)17-s + ⋯
L(s)  = 1  + (0.449 − 0.893i)2-s + (−0.900 − 0.433i)3-s + (−0.596 − 0.802i)4-s + (−0.985 − 0.171i)5-s + (−0.792 + 0.609i)6-s + (0.770 − 0.636i)7-s + (−0.985 + 0.171i)8-s + (0.623 + 0.781i)9-s + (−0.596 + 0.802i)10-s + (−0.354 − 0.935i)11-s + (0.188 + 0.982i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.813 + 0.582i)15-s + (−0.289 + 0.957i)16-s + (−0.952 + 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2186712465 - 0.1589261053i\)
\(L(\frac12)\) \(\approx\) \(-0.2186712465 - 0.1589261053i\)
\(L(1)\) \(\approx\) \(0.3950256551 - 0.5046941607i\)
\(L(1)\) \(\approx\) \(0.3950256551 - 0.5046941607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.449 - 0.893i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.985 - 0.171i)T \)
7 \( 1 + (0.770 - 0.636i)T \)
11 \( 1 + (-0.354 - 0.935i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.952 + 0.305i)T \)
19 \( 1 + (-0.928 - 0.370i)T \)
23 \( 1 + (-0.994 + 0.103i)T \)
29 \( 1 + (0.851 + 0.524i)T \)
31 \( 1 + (-0.154 - 0.987i)T \)
37 \( 1 + (-0.700 + 0.713i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.999 - 0.0345i)T \)
47 \( 1 + (0.120 + 0.992i)T \)
53 \( 1 + (0.813 + 0.582i)T \)
59 \( 1 + (0.568 + 0.822i)T \)
61 \( 1 + (-0.418 - 0.908i)T \)
67 \( 1 + (0.188 + 0.982i)T \)
71 \( 1 + (0.997 - 0.0689i)T \)
73 \( 1 + (0.813 - 0.582i)T \)
79 \( 1 + (-0.999 - 0.0345i)T \)
83 \( 1 + (-0.748 + 0.663i)T \)
89 \( 1 + (-0.418 + 0.908i)T \)
97 \( 1 + (0.990 + 0.137i)T \)
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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

−24.0017940658655586040479054956, −23.13580246467099567886790343999, −22.722882508212170897815017321066, −21.59450224484197849464789975095, −21.23564597414086515459962430704, −19.96243241697251074410271055148, −18.610606278091263922754415382882, −17.96957465210900920883456006265, −17.24222728451724262953873539126, −16.1900584354023177895208946777, −15.63897689290702041513749494879, −14.96244556667950094136974912342, −14.20200009201398658851729534206, −12.706138080325780560114508852400, −12.04837244378136043492594533042, −11.47041592597156475748657752029, −10.30602290183691353685150771038, −9.036736192461804890863246890641, −8.20276882567325158745884729617, −7.10661128026648896501806610290, −6.46985126895405664359835052050, −5.21862040201264105769280289483, −4.518443561747934170108179069064, −3.93067957922057782939008541862, −2.215859483675750682876022009482, 0.15356571163231722084032601773, 1.17038695097699482727714301954, 2.52873367404091131472459395799, 3.922171195361264876877151911216, 4.6294631399169816410824173172, 5.52753053576608279693166262403, 6.61608498785404861085890035320, 7.88705881195859454132749832200, 8.545441070665255547852584488732, 10.263948454925428809235436079805, 10.91221799677250923079859054937, 11.39626749756781265109082285862, 12.32936590621224632630672753885, 13.07887937334851632276490055816, 13.85529801152393960728283870288, 15.05661084484887393057409718201, 15.84123308914522365572839912422, 17.01016348708781668833797496780, 17.81958245562054797989893074793, 18.59041797630495775829477672390, 19.54971043494171244129815606136, 20.05953050424650187623489466796, 21.12199649566796694591452154992, 21.97219632334879855071888700864, 22.71558977463710926843463976019

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