L-function 1-675-675.302-r0-0-0

• Label: 1-675-675.302-r0-0-0
• Degree: $1$
• Conductor: $675$
• Sign: $0.861 + 0.508i$
• Analytic cond.: $3.13468$
• Root an. cond.: $3.13468$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.469 + 0.882i)2^-s + (−0.559 − 0.829i)4^-s + (−0.342 − 0.939i)7^-s + (0.994 − 0.104i)8^-s + (−0.848 + 0.529i)11^-s + (0.469 + 0.882i)13^-s + (0.990 + 0.139i)14^-s + (−0.374 + 0.927i)16^-s + (0.406 − 0.913i)17^-s + (0.104 + 0.994i)19^-s + (−0.0697 − 0.997i)22^-s + (−0.788 − 0.615i)23^-s − 26^-s + (−0.587 + 0.809i)28^-s + (−0.719 + 0.694i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$675$    =    $3^{3} \cdot 5^{2}$
Sign:	$0.861 + 0.508i$
Analytic conductor:	$3.13468$
Root analytic conductor:	$3.13468$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (302, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 675,\ (0:\ ),\ 0.861 + 0.508i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8804193334 + 0.2403050213i$
$L(1)$	$\approx$	$0.7508168775 + 0.2118892888i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
good	2	$1 + (-0.469 + 0.882i)T$
	7	$1 + (-0.342 - 0.939i)T$
	11	$1 + (-0.848 + 0.529i)T$
	13	$1 + (0.469 + 0.882i)T$
	17	$1 + (0.406 - 0.913i)T$
	19	$1 + (0.104 + 0.994i)T$
	23	$1 + (-0.788 - 0.615i)T$
	29	$1 + (-0.719 + 0.694i)T$
	31	$1 + (0.961 - 0.275i)T$

Imaginary part of the first few zeros on the critical line

−22.48473299805601344401354533869, −21.56254283163451259329121641745, −21.22537165096542969410771533277, −20.133443584826854375746038152002, −19.37557447637925845525273219984, −18.68338050197169335835340481983, −17.93880534724909680691369386478, −17.25165519074955382013755234445, −15.987869772136604971173860227412, −15.53991539333910140745485538893, −14.22130224931993291230199043471, −13.02529674563971015730701452059, −12.822560883603400056979109379525, −11.62995852091689721468178776931, −10.95180533928793509708051083701, −10.02932479217277993803989879295, −9.236236344713649251022877817168, −8.26173072810605222073687716897, −7.73419557288278817844219452005, −6.11967744668878109288352562755, −5.34034365264499161567200736848, −4.01860571607189903508479927388, −2.98537421249473807726627162425, −2.319992979152201061152045202536, −0.85895130314335339193093774411, 0.76038453361757374079247475227, 2.10705968122783193277985003339, 3.754294560978448491549654335572, 4.617646086206680755185435538954, 5.6703735295256804457724549321, 6.650878807908376686017409598274, 7.41760954366524347799709603112, 8.12867361521034270592103385573, 9.27020564396483281598015153234, 10.06146696642262629049168593704, 10.672058192516090850257781308850, 11.938968261619597635919680649713, 13.136638342964944863972491954455, 13.863424508139250828168980059924, 14.52708785644968166956482412810, 15.611292803947042901992322237492, 16.416595108146872082523966478933, 16.770432928516290198240415228311, 17.99271527184586745924346878725, 18.49675166674659074392936546443, 19.35762887437169594209783145978, 20.378226095940763670363159426628, 20.95199252408038923520517975679, 22.44800983325245484051848621657, 22.95780279304789501452645659621

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