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Properties

Label 1-539-539.207-r0-0-0
Degree $1$
Conductor $539$
Sign $0.894 + 0.446i$
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  + (−0.646 + 0.762i)2-s + (0.0149 − 0.999i)3-s + (−0.163 − 0.986i)4-s + (0.992 + 0.119i)5-s + (0.753 + 0.657i)6-s + (0.858 + 0.512i)8-s + (−0.999 − 0.0299i)9-s + (−0.733 + 0.680i)10-s + (−0.988 + 0.149i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (−0.946 + 0.323i)16-s + (0.251 + 0.967i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + (−0.0448 − 0.998i)20-s + ⋯
L(s)  = 1  + (−0.646 + 0.762i)2-s + (0.0149 − 0.999i)3-s + (−0.163 − 0.986i)4-s + (0.992 + 0.119i)5-s + (0.753 + 0.657i)6-s + (0.858 + 0.512i)8-s + (−0.999 − 0.0299i)9-s + (−0.733 + 0.680i)10-s + (−0.988 + 0.149i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (−0.946 + 0.323i)16-s + (0.251 + 0.967i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + (−0.0448 − 0.998i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.894 + 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158545852 + 0.2727947651i\)
\(L(\frac12)\) \(\approx\) \(1.158545852 + 0.2727947651i\)
\(L(1)\) \(\approx\) \(0.9462950001 + 0.1067863772i\)
\(L(1)\) \(\approx\) \(0.9462950001 + 0.1067863772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.646 + 0.762i)T \)
3 \( 1 + (0.0149 - 0.999i)T \)
5 \( 1 + (0.992 + 0.119i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (0.251 + 0.967i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.447 - 0.894i)T \)
41 \( 1 + (0.858 + 0.512i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.925 - 0.379i)T \)
53 \( 1 + (0.712 + 0.701i)T \)
59 \( 1 + (-0.873 - 0.486i)T \)
61 \( 1 + (0.712 - 0.701i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.925 + 0.379i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
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

−22.73732694050750857034208789200, −22.43711033901830868058778566147, −21.41818230610209138651130981378, −20.67183042429020032216485622728, −20.49030116023157259543758508578, −19.20501701635690686238097145943, −18.18780174694109783306960768035, −17.600463208841555711472622441199, −16.60207256922932254916324069826, −16.116476272165534159592387531473, −14.90131727854874190646916455215, −13.71834753968028587686309784194, −13.203712509303050177373067855563, −11.830056495606379207150726711, −11.072828053846286476878603337363, −10.26904571478094503703315944611, −9.4595815087172149617350448149, −8.96506149162050452809364993985, −7.900983006371626097355939412735, −6.48201212118215770565871902115, −5.31741737927258522395455361589, −4.36290392378644895292489199263, −3.17577728817219862534061499847, −2.39524003130304447204594176244, −0.909712663387754895830719738912, 1.2993073813234372235651524335, 1.81839558286320759417197246223, 3.41295885263853615898885602307, 5.32807513775589005696572501756, 5.894498273939684654070355249422, 6.71814964124874515905537317002, 7.57665742576691361825245029291, 8.5757262379208046041630591033, 9.267081027270258770756649141457, 10.40283306768215524042869577823, 11.191098211448289965716699538789, 12.54866238411251630189394312923, 13.46086710676355625921590603491, 14.10774619278900160302401305977, 14.83931893941809731547798388035, 16.11299241123393923158568107360, 16.9265147441954487210624370863, 17.71360903820233699797586575967, 18.26990652888297344732737442191, 18.97349348631841145532039167701, 19.859162638646106789764999826966, 20.83394464244003496874877116921, 21.94013898909982579526134675765, 23.11653224651034039099071780731, 23.53161417513341193065617690292

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