L-function 1-547-547.119-r0-0-0

• Label: 1-547-547.119-r0-0-0
• Degree: $1$
• Conductor: $547$
• Sign: $0.534 + 0.845i$
• Analytic cond.: $2.54025$
• Root an. cond.: $2.54025$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.976 + 0.216i)2^-s + (−0.900 − 0.433i)3^-s + (0.905 − 0.423i)4^-s + (−0.131 + 0.991i)5^-s + (0.973 + 0.228i)6^-s + (0.863 − 0.504i)7^-s + (−0.792 + 0.609i)8^-s + (0.623 + 0.781i)9^-s + (−0.0862 − 0.996i)10^-s + (−0.919 − 0.391i)11^-s + (−0.999 − 0.0115i)12^-s + (0.955 + 0.294i)13^-s + (−0.733 + 0.680i)14^-s + (0.548 − 0.835i)15^-s + (0.641 − 0.767i)16^-s + (0.0287 + 0.999i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$547$
Sign:	$0.534 + 0.845i$
Analytic conductor:	$2.54025$
Root analytic conductor:	$2.54025$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{547} (119, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 547,\ (0:\ ),\ 0.534 + 0.845i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5709361684 + 0.3145520072i$
$L(1)$	$\approx$	$0.5843362288 + 0.1054940408i$

Euler product

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

	$p$	$F_p(T)$
bad	547	$1$
good	2	$1 + (-0.976 + 0.216i)T$
	3	$1 + (-0.900 - 0.433i)T$
	5	$1 + (-0.131 + 0.991i)T$
	7	$1 + (0.863 - 0.504i)T$
	11	$1 + (-0.919 - 0.391i)T$
	13	$1 + (0.955 + 0.294i)T$
	17	$1 + (0.0287 + 0.999i)T$
	19	$1 + (0.999 - 0.0230i)T$
	23	$1 + (-0.332 + 0.942i)T$

Imaginary part of the first few zeros on the critical line

−23.37249798700030563522059453834, −22.309836360399261620724537306846, −21.12775679897024215587332720166, −20.79511075841950406885743882775, −20.197528445948040781818662336128, −18.67438070549624265732018801557, −18.08014504026461998805969785852, −17.5294806170250423880301873447, −16.52352979586295798000852387236, −15.7583084181217224776429503238, −15.46892121958841903957434514168, −13.75214731944932602268092772783, −12.358943638383855189057847944, −12.04858171200633727691254446987, −11.07346297825613890657459967223, −10.30130969552484060246356871953, −9.34744678534856016182699736408, −8.47243831049768278371820455810, −7.71678813661660948116833147992, −6.448818584415688542414323841237, −5.30219789974494816216262017292, −4.70211644320926218664224718141, −3.182196360657523213872036839261, −1.6984829460274084569778181379, −0.65888566327619274384804287077, 1.085806896620298234958073416244, 2.075338135908949861232538497396, 3.50627784654027667445477839226, 5.113379873144603833760164576823, 6.08327541862886246210528254321, 6.81868853350955885890567822519, 7.81999436016976662034691679669, 8.24055868147892386096927089363, 10.01725960168282885159859996689, 10.54257358785591698247851019898, 11.35030566631579681282353964293, 11.78366982707194571147375923468, 13.39282380931418260835190225136, 14.15809244800294244262136238030, 15.43739094950054267489575842211, 15.944781488704054616540025836114, 17.0699611201240574272976446508, 17.75470484573157435332500995415, 18.33560019460308217424268785122, 18.95395224862136819458310250675, 19.88188905611587524679485401406, 21.11212979398273217559017808588, 21.67633803583242879613329689470, 23.11601142782632528579809492056, 23.52955178747325394513187225372

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