L-function 1-1800-1800.533-r0-0-0

• Label: 1-1800-1800.533-r0-0-0
• Degree: $1$
• Conductor: $1800$
• Sign: $-0.916 - 0.400i$
• Analytic cond.: $8.35916$
• Root an. cond.: $8.35916$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 + 0.5i)7^-s + (0.913 + 0.406i)11^-s + (−0.406 − 0.913i)13^-s + (−0.951 + 0.309i)17^-s + (0.309 + 0.951i)19^-s + (−0.994 + 0.104i)23^-s + (0.978 − 0.207i)29^-s + (−0.978 − 0.207i)31^-s + (0.587 − 0.809i)37^-s + (−0.913 + 0.406i)41^-s + (−0.866 + 0.5i)43^-s + (−0.207 − 0.978i)47^-s + (0.5 − 0.866i)49^-s + (0.951 + 0.309i)53^-s + (−0.913 + 0.406i)59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1800$    =    $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.916 - 0.400i$
Analytic conductor:	$8.35916$
Root analytic conductor:	$8.35916$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (533, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1800,\ (0:\ ),\ -0.916 - 0.400i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.01821360798 - 0.08718362399i$
$L(1)$	$\approx$	$0.7768512248 + 0.04057857096i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1$
good	7	$1 + (-0.866 + 0.5i)T$
	11	$1 + (0.913 + 0.406i)T$
	13	$1 + (-0.406 - 0.913i)T$
	17	$1 + (-0.951 + 0.309i)T$
	19	$1 + (0.309 + 0.951i)T$
	23	$1 + (-0.994 + 0.104i)T$
	29	$1 + (0.978 - 0.207i)T$
	31	$1 + (-0.978 - 0.207i)T$
	37	$1 + (0.587 - 0.809i)T$

Imaginary part of the first few zeros on the critical line

−20.04262028328982029032844459784, −20.00052986313866687513969422657, −19.16901768775666979523176864875, −18.34957828263001729622130698515, −17.49748786778897665866618397664, −16.7538210511278644365060612166, −16.19830460733234878184453669072, −15.46797303921547229625921334277, −14.471495034197165472237325638056, −13.75877911677391204759327410453, −13.28391316178291896840935777368, −12.188758653495869856234995688181, −11.63362549928744006564165945932, −10.77836686492040468025498735112, −9.87649007984227046767674254103, −9.20735228253521492446660698547, −8.58821497131260930588988180623, −7.33674126806694326900945073517, −6.7057311575465062681584268673, −6.19070376484107263608869170034, −4.89749339374391682989505597670, −4.17049968899917446178297714866, −3.34025837824566655591982472881, −2.38676243395002914970236697474, −1.26828126876440409836568455908, 0.031367621228065306024729233575, 1.54945204285917583171935493470, 2.46682915415177816566459204601, 3.44580727836090186451572894791, 4.1688125644052255920238933643, 5.275923721086239778285965910127, 6.11197134574765969413198205528, 6.70112867863060703467247783007, 7.70854429024353612977880870246, 8.52189210079761378516233626984, 9.40985431488211004007568394528, 9.963386998108973456465548981671, 10.76964435490155694283807959341, 11.94189578557000306750857650480, 12.27827380057505682597795897765, 13.13653814336921165837567160839, 13.89011071756739865593758560655, 14.92514348035875340397341507731, 15.28473289309303850360963298766, 16.30041967375302641937854273012, 16.7932503301040042388298746879, 17.86588342757115182376955178126, 18.23700527230334048320940291235, 19.30043608022803010809077455372, 19.97236623115307685855769066160

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