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Properties

Label 1-539-539.101-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.485 - 0.874i$
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  + (−0.193 − 0.981i)2-s + (0.447 − 0.894i)3-s + (−0.925 + 0.379i)4-s + (0.842 + 0.538i)5-s + (−0.963 − 0.266i)6-s + (0.550 + 0.834i)8-s + (−0.599 − 0.800i)9-s + (0.365 − 0.930i)10-s + (−0.0747 + 0.997i)12-s + (0.753 − 0.657i)13-s + (0.858 − 0.512i)15-s + (0.712 − 0.701i)16-s + (−0.999 + 0.0299i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + (−0.983 − 0.178i)20-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s + (0.447 − 0.894i)3-s + (−0.925 + 0.379i)4-s + (0.842 + 0.538i)5-s + (−0.963 − 0.266i)6-s + (0.550 + 0.834i)8-s + (−0.599 − 0.800i)9-s + (0.365 − 0.930i)10-s + (−0.0747 + 0.997i)12-s + (0.753 − 0.657i)13-s + (0.858 − 0.512i)15-s + (0.712 − 0.701i)16-s + (−0.999 + 0.0299i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + (−0.983 − 0.178i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.485 - 0.874i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.485 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7816463376 - 1.328445832i\)
\(L(\frac12)\) \(\approx\) \(0.7816463376 - 1.328445832i\)
\(L(1)\) \(\approx\) \(0.9169769554 - 0.7725585085i\)
\(L(1)\) \(\approx\) \(0.9169769554 - 0.7725585085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.193 - 0.981i)T \)
3 \( 1 + (0.447 - 0.894i)T \)
5 \( 1 + (0.842 + 0.538i)T \)
13 \( 1 + (0.753 - 0.657i)T \)
17 \( 1 + (-0.999 + 0.0299i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (0.691 + 0.722i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.971 - 0.237i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.873 - 0.486i)T \)
53 \( 1 + (0.525 - 0.850i)T \)
59 \( 1 + (-0.998 - 0.0598i)T \)
61 \( 1 + (0.525 + 0.850i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (-0.873 - 0.486i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.753 + 0.657i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
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

−23.80297762812882224040133974515, −22.95606746256097203101837640812, −21.78719663992772389533852865605, −21.54018502633537810077730217954, −20.35415075151233758359240691089, −19.588450517724941522527678741365, −18.42942928855107666804139049411, −17.538309439985330303523384229083, −16.85078394284166695418145515679, −15.97912058202753569590570889108, −15.46246647902497742050149515147, −14.37815206082876403307477311528, −13.62520150621344978180425224752, −13.14321011893190715075003525979, −11.42716147806597097738943214841, −10.3683559839756183314650368341, −9.41057839833407530695755851808, −8.98660204846491107955915833114, −8.16216912660371324258119192488, −6.83609321854881574824448449974, −5.906484612507051694487103715238, −4.89791147979001699462249983618, −4.31693529083851879049437620504, −2.87903808982530820522635540009, −1.33555054033001712833281224084, 0.967976322022990119201763646845, 2.02556493773196265130161926395, 2.826039251145120432970052496451, 3.73790395206705259431725923782, 5.31202528295295231882229612964, 6.33097228326469667005478660403, 7.39315932972162423403532710400, 8.48011952028962110876810632472, 9.158194836946127405878634400012, 10.23156190740596441896937240955, 11.00619277796924002915525547908, 11.99390893672338481280004791456, 13.005381663569773646633293838730, 13.48587349783939977317342260665, 14.25007089577675710038423394888, 15.191948928956086748974080981126, 16.81525597952439227767051836099, 17.67622685791943525658107460401, 18.35390742800448337788752303787, 18.74503680676643305205535803381, 19.90378933801161344010295564686, 20.49595317279290210630456082927, 21.2714246191499324230557973254, 22.35874093908345037624787170328, 22.86711663496312830325338970044

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