L-function 1-1045-1045.189-r0-0-0

• Label: 1-1045-1045.189-r0-0-0
• Degree: $1$
• Conductor: $1045$
• Sign: $0.0219 + 0.999i$
• Analytic cond.: $4.85295$
• Root an. cond.: $4.85295$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.809 + 0.587i)2^-s + (0.309 − 0.951i)3^-s + (0.309 + 0.951i)4^-s + (0.809 − 0.587i)6^-s + (0.309 + 0.951i)7^-s + (−0.309 + 0.951i)8^-s + (−0.809 − 0.587i)9^-s + 12^-s + (0.809 + 0.587i)13^-s + (−0.309 + 0.951i)14^-s + (−0.809 + 0.587i)16^-s + (−0.809 + 0.587i)17^-s + (−0.309 − 0.951i)18^-s + 21^-s − 23^-s + (0.809 + 0.587i)24^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1045$    =    $5 \cdot 11 \cdot 19$
Sign:	$0.0219 + 0.999i$
Analytic conductor:	$4.85295$
Root analytic conductor:	$4.85295$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1045} (189, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1045,\ (0:\ ),\ 0.0219 + 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.758066028 + 1.719832472i$
$L(1)$	$\approx$	$1.629514152 + 0.6145153309i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	11	$1$
	19	$1$
good	2	$1 + (0.809 + 0.587i)T$
	3	$1 + (0.309 - 0.951i)T$
	7	$1 + (0.309 + 0.951i)T$
	13	$1 + (0.809 + 0.587i)T$
	17	$1 + (-0.809 + 0.587i)T$
	23	$1 - T$
	29	$1 + (0.309 + 0.951i)T$
	31	$1 + (0.809 + 0.587i)T$
	37	$1 + (0.309 + 0.951i)T$

Imaginary part of the first few zeros on the critical line

−21.183647824923817033779170473126, −20.700610911737716766348731074718, −20.01962463885655597000524105671, −19.53244802751086340491574025390, −18.29911348820409810824319538820, −17.44135030534633101442622863299, −16.333515813303329204140793321686, −15.70646525710010127992495083957, −15.02810981313017565889037045621, −13.992004642671172835342167819678, −13.72941288638916370910754072371, −12.79280737305743668250719559967, −11.476757777844845053238266620101, −11.109521943891513868498696051318, −10.20803529648277056158540909510, −9.684095477996386263350512564723, −8.542804725397515278744260677932, −7.584658580766517080379044654680, −6.33121846304769382243790523795, −5.52206545044516520456675116095, −4.37821695324002447874244229085, −4.11811204978946636973566824357, −3.0604123167326531945428755504, −2.15453409042674043502035412178, −0.73017361818717432113446046031, 1.61036198836225163651328647215, 2.401738086255448559506162587656, 3.36674553449239030678131716305, 4.41747545748980578920344753649, 5.50760207971898920559518940119, 6.32011558021839988908850446920, 6.79544889630490647601635609885, 8.07250123276875296713046323226, 8.42967739410292905320145081636, 9.2858002345733657618216001133, 11.00578554845193018925803503658, 11.667678791477422776129531541238, 12.48879956758018076913166487669, 12.9855306423227929744503396662, 14.129555613207375350235554753951, 14.27765387253752607822692027724, 15.51219645817455272256740864449, 15.90058213982295355323801375917, 17.16492908340243969426842308586, 17.80660288004752909779486944980, 18.496539520906940102647418513018, 19.37116469871119597733434840862, 20.31245660336784542343775679913, 21.05425690868370574967548733104, 21.86523547496070693570958213708

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