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Properties

Label 1-539-539.237-r0-0-0
Degree $1$
Conductor $539$
Sign $0.772 + 0.635i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.772 + 0.635i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.772 + 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255021924 + 0.4501510118i\)
\(L(\frac12)\) \(\approx\) \(1.255021924 + 0.4501510118i\)
\(L(1)\) \(\approx\) \(1.026109875 + 0.3428238774i\)
\(L(1)\) \(\approx\) \(1.026109875 + 0.3428238774i\)

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

−23.1244525383331775772418137502, −22.26728413405571303580267367132, −21.55636493994381178463017590625, −21.28350325241382159783500839851, −20.164051006765307503961238639082, −19.10454601826290081346326128602, −18.2121123257438907916163238257, −17.65678705370656160909881480913, −16.73430608643024342824913668957, −15.70626901443953038076041153367, −14.4967135902992490528919732489, −13.92876692710445204522760046504, −12.848531241658856820553854193338, −12.0566044127593558094002724238, −11.25991783706966871652350681636, −10.41152154570040701285644390823, −9.69187041426328645280823205140, −9.04792348471082185585374052540, −7.152209206797776678767049827839, −6.12954539696103978805062750366, −5.35413781663162475725656744065, −4.55762460731567063514176200843, −3.38470034780388093133275038841, −2.17480406458432007899914292247, −1.06367970669374951145886139928, 0.91422902936563219790820747545, 2.4704834068907452566926375819, 4.014248731650982727941193514688, 5.15619495794831500590734375868, 5.77403020667526906747095420916, 6.37756689940632692820409701544, 7.5885368365854829474106857162, 8.26944000710428604042469268035, 9.79476159286351414104397734214, 10.23774005625261101269519469425, 11.86970493701242279895862164183, 12.46722971453522961102434910095, 13.30847106069831561088150142222, 14.0374896241329175110210606144, 15.05370561401715647535363027777, 16.09627001073774543717845372469, 16.8916889762742469666500487426, 17.337838002146435184982915326001, 18.21066509783737077233470967728, 18.84399496228549887294758198144, 20.47585416341309811390850496380, 21.312908609171365690429102774758, 22.15182745644416276585744727805, 22.717214193256630748232845887755, 23.509775013694854769931637929224

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