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Properties

Label 1-539-539.247-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.931 - 0.362i$
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  + (−0.999 + 0.0299i)2-s + (0.420 − 0.907i)3-s + (0.998 − 0.0598i)4-s + (−0.946 − 0.323i)5-s + (−0.393 + 0.919i)6-s + (−0.995 + 0.0896i)8-s + (−0.646 − 0.762i)9-s + (0.955 + 0.294i)10-s + (0.365 − 0.930i)12-s + (0.473 − 0.880i)13-s + (−0.691 + 0.722i)15-s + (0.992 − 0.119i)16-s + (0.887 + 0.460i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (−0.963 − 0.266i)20-s + ⋯
L(s)  = 1  + (−0.999 + 0.0299i)2-s + (0.420 − 0.907i)3-s + (0.998 − 0.0598i)4-s + (−0.946 − 0.323i)5-s + (−0.393 + 0.919i)6-s + (−0.995 + 0.0896i)8-s + (−0.646 − 0.762i)9-s + (0.955 + 0.294i)10-s + (0.365 − 0.930i)12-s + (0.473 − 0.880i)13-s + (−0.691 + 0.722i)15-s + (0.992 − 0.119i)16-s + (0.887 + 0.460i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (−0.963 − 0.266i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.931 - 0.362i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.931 - 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1145848634 - 0.6106365650i\)
\(L(\frac12)\) \(\approx\) \(0.1145848634 - 0.6106365650i\)
\(L(1)\) \(\approx\) \(0.5493647084 - 0.3360372723i\)
\(L(1)\) \(\approx\) \(0.5493647084 - 0.3360372723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 + 0.0299i)T \)
3 \( 1 + (0.420 - 0.907i)T \)
5 \( 1 + (-0.946 - 0.323i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (0.887 + 0.460i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.936 - 0.351i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.772 - 0.635i)T \)
41 \( 1 + (-0.995 + 0.0896i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.280 + 0.959i)T \)
53 \( 1 + (-0.842 - 0.538i)T \)
59 \( 1 + (0.575 - 0.817i)T \)
61 \( 1 + (-0.842 + 0.538i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (-0.280 - 0.959i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.473 + 0.880i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
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

−23.812916760651614455637261862388, −23.00548628670306866460724963589, −21.8858756529371218771513204694, −21.07343219810977792316894798352, −20.28801591578569615807929044310, −19.62615636067896719899967620631, −18.82169507437075427790717649192, −18.10851441634626346277674396252, −16.75370275148803431145088852349, −16.15946136490687640168479210786, −15.66163892492646220458688977802, −14.62172749342642176060974337881, −13.94194766861238303698789152658, −12.05822713223583845718034409882, −11.65723480525323077567355799634, −10.54334404032326097306269460979, −9.972954034966639324451822388367, −8.916413132376068166137464358657, −8.19117478061252437556680630384, −7.40454370895999143729780553181, −6.27932621895131791346939470488, −4.91677029638307121869447711574, −3.62205474835744386534179207683, −3.04586509062377213148788221184, −1.55180562073559828137507843188, 0.454007789795836897519374964695, 1.487694916920306257773026701924, 2.823472252687966773948718349841, 3.68598068641047396031851303517, 5.4889213966520981578498880565, 6.52759227290839521408960348953, 7.56006718056836532043727915027, 8.06471573986596693489256843747, 8.75968121975406171864485005206, 9.86605012203148229446444117647, 10.98112995540645145934502133447, 11.92601852040181141675176957862, 12.442053948161590227880965442449, 13.57270195175668374512599465796, 14.7683034547189755708429468211, 15.531742005993411097691090967553, 16.324584470360737249103323292685, 17.41375265630401549351606497746, 18.0613445220217535062185093850, 18.99122500684204616500743216652, 19.48504112213807851599288104569, 20.35035660523257664520929478129, 20.78181097471124199048289939090, 22.34173182869185045844080100870, 23.517084182515959690053553531733

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