L-function 1-23-23.6-r0-0-0

• Label: 1-23-23.6-r0-0-0
• Degree: $1$
• Conductor: $23$
• Sign: $0.854 - 0.519i$
• Analytic cond.: $0.106811$
• Root an. cond.: $0.106811$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.654 − 0.755i)2^-s + (0.841 + 0.540i)3^-s + (−0.142 + 0.989i)4^-s + (0.415 − 0.909i)5^-s + (−0.142 − 0.989i)6^-s + (−0.959 − 0.281i)7^-s + (0.841 − 0.540i)8^-s + (0.415 + 0.909i)9^-s + (−0.959 + 0.281i)10^-s + (−0.654 + 0.755i)11^-s + (−0.654 + 0.755i)12^-s + (−0.959 + 0.281i)13^-s + (0.415 + 0.909i)14^-s + (0.841 − 0.540i)15^-s + (−0.959 − 0.281i)16^-s + (−0.142 − 0.989i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$23$
Sign:	$0.854 - 0.519i$
Analytic conductor:	$0.106811$
Root analytic conductor:	$0.106811$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{23} (6, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 23,\ (0:\ ),\ 0.854 - 0.519i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5936953919 - 0.1662737264i$
$L(1)$	$\approx$	$0.8044174782 - 0.1800159172i$

Euler product

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

	$p$	$F_p(T)$
bad	23	$1$
good	2	$1 + (-0.654 - 0.755i)T$
	3	$1 + (0.841 + 0.540i)T$
	5	$1 + (0.415 - 0.909i)T$
	7	$1 + (-0.959 - 0.281i)T$
	11	$1 + (-0.654 + 0.755i)T$
	13	$1 + (-0.959 + 0.281i)T$
	17	$1 + (-0.142 - 0.989i)T$
	19	$1 + (-0.142 + 0.989i)T$
	29	$1 + (-0.142 - 0.989i)T$

Imaginary part of the first few zeros on the critical line

−38.700408365765894581394546516791, −37.374582052745730403293364755146, −36.73684444152570151931867284665, −35.19931983131626181222038711038, −34.39659421649262485665831178230, −32.686616841364382480753785584274, −31.68697678664476224776710908677, −29.90051658251202674860051032003, −28.79736871063084622912026153343, −26.68424173530906038326597384210, −26.03410951890411644829689228165, −24.959998697416946641111554588972, −23.61823174774936248044005691048, −21.899117064346872748693005649887, −19.63149511760766360944426352873, −18.86079323503127356346653185205, −17.60123837809627875346084807219, −15.65730440476089251256326484275, −14.4875947273727442066818885853, −13.12631838826265572682657003615, −10.40381591166334108468080526150, −9.0236073970210270980590936589, −7.386736091739146298255844220707, −6.13271706638252198914368019481, −2.695063685459656135537545911260, 2.47039178966832528664961712539, 4.432117621973441540755241160510, 7.720799325920089958960128054403, 9.39325673406057682914387718310, 10.03959677418238791346100433673, 12.425109771331570931137449711022, 13.61944875373012828405910231195, 15.89863210493409896148305765933, 17.07638010143655944598506026097, 18.98899807308690491928557869720, 20.21715299352348244292115288302, 20.977937859874896897254665650164, 22.4678459530138552535662630855, 24.93649520853625830521165936466, 25.9574329561884069176049822057, 27.13223980683776795140294166042, 28.50879775493850333463910445784, 29.53655072465957060785288875489, 31.32056718767089627329744047203, 32.1091735901850118059187416565, 33.703928987552825404236293326258, 35.938907075447944483317436388978, 36.27341873101567924360620203011, 37.635050660222910611763365632394, 38.82904969900439216797491855283

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