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Properties

Label 1-539-539.307-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.914 - 0.404i$
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  + (0.900 − 0.433i)2-s + (0.222 − 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)6-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 18-s + 19-s + (−0.623 − 0.781i)20-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.222 − 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)6-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 18-s + 19-s + (−0.623 − 0.781i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.914 - 0.404i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5104995529 - 2.414394773i\)
\(L(\frac12)\) \(\approx\) \(0.5104995529 - 2.414394773i\)
\(L(1)\) \(\approx\) \(1.238020122 - 1.375696215i\)
\(L(1)\) \(\approx\) \(1.238020122 - 1.375696215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 - T \)
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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

−23.51624275806585429562441177278, −22.76430123497759166842612840216, −22.0551119725750068966978431940, −21.68285213677968385863322010951, −20.57942996619918185260085436981, −20.00732321843212699670303418972, −18.7727224277830860658636504045, −17.59638783865447478798500131353, −16.84849373734286493397337102990, −15.84967583031898246745844375269, −15.25771416235575913580249852039, −14.31244833443534114997338242966, −14.06720026888699887667324369491, −12.78898268841055821473004268947, −11.63782282019340084792972340000, −10.95719377730197227678015205876, −9.968246953591038049410113303649, −9.07007074894966443042120138009, −7.57765913677172705086325189516, −7.15571422863104519982895770762, −5.55538665628071966015754039022, −5.293228558968270595701825275539, −3.85489864193382180682044143092, −3.154350782395528616096123372924, −2.324202649440059958895356219683, 0.94956952402350678237884787572, 1.87774194622425303922299987890, 2.87783297800609022174314649520, 4.1285489747342377402450876615, 5.22040981008867518019757955420, 5.96409991429598574764211441816, 7.04978548914588758967752213281, 7.97408724792590118771361106053, 9.17107362845145243182408684785, 10.01970967864435185535043854793, 11.44658605537505391386652602492, 12.07240990777894544902097806567, 12.86547570110393342140005605357, 13.37316238012571622590451937603, 14.38975298361385969192619780613, 14.99503066857085083371928311266, 16.41110085613562558097414727397, 17.00395778175065174753622191183, 18.23435214493603545227609284058, 19.11974424860822949147331451524, 19.862487872867237101966052452186, 20.49546288726641970088145536924, 21.32356601082284792899969065004, 22.24856186632167043071010530105, 23.20779837242988948182755835136

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