L-function 1-3895-3895.113-r0-0-0

• Label: 1-3895-3895.113-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $-0.989 - 0.141i$
• Analytic cond.: $18.0883$
• Root an. cond.: $18.0883$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.951 + 0.309i)2^-s + i·3^-s + (0.809 − 0.587i)4^-s + (−0.309 − 0.951i)6^-s + (0.951 + 0.309i)7^-s + (−0.587 + 0.809i)8^-s − 9^-s + (0.809 + 0.587i)11^-s + (0.587 + 0.809i)12^-s + (−0.951 + 0.309i)13^-s − 14^-s + (0.309 − 0.951i)16^-s + (0.587 − 0.809i)17^-s + (0.951 − 0.309i)18^-s + (−0.309 + 0.951i)21^-s + (−0.951 − 0.309i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$-0.989 - 0.141i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (113, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ -0.989 - 0.141i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.05200015188 + 0.7333831590i$
$L(1)$	$\approx$	$0.5879420060 + 0.3908900659i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
	41	$1$
good	2	$1 + (-0.951 + 0.309i)T$
	3	$1 + iT$
	7	$1 + (0.951 + 0.309i)T$
	11	$1 + (0.809 + 0.587i)T$
	13	$1 + (-0.951 + 0.309i)T$
	17	$1 + (0.587 - 0.809i)T$
	23	$1 + (-0.951 + 0.309i)T$
	29	$1 + (0.809 - 0.587i)T$
	31	$1 + (0.809 + 0.587i)T$

Imaginary part of the first few zeros on the critical line

−18.102844232580913728221126398196, −17.60312106953656940807966764097, −16.98021861955808965358688584639, −16.65776348998138593204132193245, −15.48617008240154837098059979442, −14.61148958673430880439559754747, −14.179100819430835913308858165389, −13.26586908178788885843903993787, −12.40485351466827625005812805940, −11.85000337460914216545670790496, −11.46355767855421010826889970503, −10.482446789939241453082653175617, −9.98306289696234226777590302581, −8.88107590764575231562336367426, −8.22616879221748310117112431799, −7.931935911162578629822548198293, −7.04623284735927093486744421422, −6.43402601673676709697371863941, −5.65604071247792353172460095723, −4.557298821972133717305115410031, −3.484572006268559098501401323463, −2.73386039139824691067573756140, −1.73662286826529842894111711386, −1.34270511155987952531896116439, −0.2961505980452717329150567932, 1.15399025328717130860373637382, 2.10654986589800086457361121658, 2.80148833530968293336428480887, 3.95549978070035477193262362990, 4.87390423017895542617951433145, 5.267257663389797186880291321339, 6.27208657809822585739423757455, 7.04708876731380277995574506049, 7.92900629691840603459438793791, 8.4476202193987432997343669351, 9.30191781530897992531811780358, 9.805373159937960395287499912816, 10.28643823917971162392602565837, 11.29371400535651646722269777292, 11.810835277656462650300151902841, 12.190217090320786899869120656001, 13.918659859612870504575139139080, 14.45336516004186402924927497377, 14.80147164740296888718857177989, 15.67612932836236248651167448088, 16.1448653053870488279810116332, 16.94244688279154234104320258739, 17.5729269829017230997099212295, 17.83996097612883694930075808400, 18.90132247774901001490786083448

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