L-function 1-143-143.125-r1-0-0

• Label: 1-143-143.125-r1-0-0
• Degree: $1$
• Conductor: $143$
• Sign: $-0.0439 + 0.999i$
• Analytic cond.: $15.3674$
• Root an. cond.: $15.3674$
• 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 + (−0.809 − 0.587i)3^-s + (0.809 − 0.587i)4^-s + (0.951 + 0.309i)5^-s + (0.951 + 0.309i)6^-s + (0.587 + 0.809i)7^-s + (−0.587 + 0.809i)8^-s + (0.309 + 0.951i)9^-s − 10^-s − 12^-s + (−0.809 − 0.587i)14^-s + (−0.587 − 0.809i)15^-s + (0.309 − 0.951i)16^-s + (−0.309 + 0.951i)17^-s + (−0.587 − 0.809i)18^-s + (0.587 − 0.809i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$143$    =    $11 \cdot 13$
Sign:	$-0.0439 + 0.999i$
Analytic conductor:	$15.3674$
Root analytic conductor:	$15.3674$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{143} (125, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 143,\ (1:\ ),\ -0.0439 + 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.6030281898 + 0.6301198051i$
$L(1)$	$\approx$	$0.6554356846 + 0.1655945132i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−27.84193416148050731957131940173, −26.92183139273932931815241143649, −26.21828472349966417470687676238, −24.93720795394404242567430509958, −24.04810297268173558054166560471, −22.63463478364822113060668793073, −21.58004338063174235180256826412, −20.741402713383103986853752726781, −20.11200182809155796757345906140, −18.215797432460128360385168884337, −17.85676467151338856011511595824, −16.69400284096234057804791746015, −16.2869469020096097485179090675, −14.69110202811750068960356310239, −13.26798921757554189021617496071, −11.9150284450274839523721402057, −11.005278772229573161761037669757, −10.00135162265947356184183161867, −9.35422854504853018049441863952, −7.819115491820514524363244408782, −6.52054200166846218822940076422, −5.28316692419045598841060918684, −3.81249755807397695066653404051, −1.86874005978836914630600586010, −0.52524816686750850569916208444, 1.42435370015518509842090149936, 2.3525021082629669655646390199, 5.25632971092834227883177459388, 6.01242956789348949217307105564, 7.04504011751636420741448999450, 8.2914378064237194636941979519, 9.49367370958892972457938115908, 10.69221026612034960783329403824, 11.4868408968552190975414198468, 12.74420324944730926050694834290, 14.144501670333467695094774054907, 15.28676418357092309206707512560, 16.52325648513174407080852451117, 17.53315038198437075884076247908, 18.07910324648910341007414734351, 18.80597974346246007560672828622, 20.08973235260731701429235613456, 21.51817725853659004687775594858, 22.227611475590799127771387591425, 23.85054706098660669300310855370, 24.40573190420205976679787818403, 25.36901458329472294942833927853, 26.20592293350886365181892161978, 27.57355481909862308674415442677, 28.33508244442482684683268133152

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