L-function 1-539-539.405-r1-0-0

• Label: 1-539-539.405-r1-0-0
• Degree: $1$
• Conductor: $539$
• Sign: $-0.446 + 0.894i$
• Analytic cond.: $57.9235$
• Root an. cond.: $57.9235$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.983 + 0.178i)2^-s + (−0.858 − 0.512i)3^-s + (0.936 + 0.351i)4^-s + (0.393 + 0.919i)5^-s + (−0.753 − 0.657i)6^-s + (0.858 + 0.512i)8^-s + (0.473 + 0.880i)9^-s + (0.222 + 0.974i)10^-s + (−0.623 − 0.781i)12^-s + (−0.983 − 0.178i)13^-s + (0.134 − 0.990i)15^-s + (0.753 + 0.657i)16^-s + (0.963 + 0.266i)17^-s + (0.309 + 0.951i)18^-s + (−0.309 + 0.951i)19^-s + (0.0448 + 0.998i)20^-s + ⋯

Functional equation

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

Invariants

Degree:	$1$
Conductor:	$539$    =    $7^{2} \cdot 11$
Sign:	$-0.446 + 0.894i$
Analytic conductor:	$57.9235$
Root analytic conductor:	$57.9235$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{539} (405, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 539,\ (1:\ ),\ -0.446 + 0.894i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.346547007 + 2.175969583i$
$L(1)$	$\approx$	$1.459199820 + 0.5020306605i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	7	$1$
	11	$1$
good	2	$1 + (0.983 + 0.178i)T$
	3	$1 + (-0.858 - 0.512i)T$
	5	$1 + (0.393 + 0.919i)T$
	13	$1 + (-0.983 - 0.178i)T$
	17	$1 + (0.963 + 0.266i)T$
	19	$1 + (-0.309 + 0.951i)T$
	23	$1 + (0.623 - 0.781i)T$
	29	$1 + (-0.550 + 0.834i)T$
	31	$1 + (0.809 + 0.587i)T$

Imaginary part of the first few zeros on the critical line

−22.90534026485314879670047180002, −22.055003403957530386067375971943, −21.30873599539533332816620700538, −20.876944746070402831176781703007, −19.86109510592470233452475160051, −18.93992992625732506146761270981, −17.41014914014284884711869004555, −17.00200536581965918699153503966, −16.08429351500396584234380697565, −15.35179939481119417519084876750, −14.42273622660806122527187864441, −13.32454941181608402136902899488, −12.60142182193704400377168738489, −11.830235911004847910157663970866, −11.12289210420463559279816592626, −9.89690128874655167150784010262, −9.433617971085440835782890343139, −7.75411207987424502126416715331, −6.640447247292639930621449294765, −5.681036019268857227128884932103, −4.96545435945845735831261484038, −4.358288067275321828865755332925, −3.07174047078718460610834745831, −1.68007999428465602703969638279, −0.478457195252746224300661860224, 1.490500010442689688255623720457, 2.52440930929421296631847316828, 3.60028888697700240959198989473, 4.987180371793838065101342614760, 5.63573750382822237475658872063, 6.64056368054409970874306781611, 7.15744758245205210749509059745, 8.171789104014698429658386961550, 10.18723685093095084697622364432, 10.53018554756237314315300046298, 11.7136976568836464786005483744, 12.32207904487613617433244153829, 13.14872349951827744806892946134, 14.18198013882022190552919061568, 14.73400944877815174432749146555, 15.76585842548452697231651773499, 16.96816045143528120344739142768, 17.188915313112296644738844755290, 18.586087505693663972146679673644, 19.0642327030143094538719999623, 20.32468918091921370516696022966, 21.40839924517112029064351319302, 21.99752521533069378892060885379, 22.74970210376484046774736221151, 23.25826590423104449689545294542

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