L-function 1-143-143.74-r1-0-0

• Label: 1-143-143.74-r1-0-0
• Degree: $1$
• Conductor: $143$
• Sign: $0.735 + 0.677i$
• 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

+ (0.978 − 0.207i)2^-s + (0.913 + 0.406i)3^-s + (0.913 − 0.406i)4^-s + (0.309 + 0.951i)5^-s + (0.978 + 0.207i)6^-s + (−0.913 + 0.406i)7^-s + (0.809 − 0.587i)8^-s + (0.669 + 0.743i)9^-s + (0.5 + 0.866i)10^-s + 12^-s + (−0.809 + 0.587i)14^-s + (−0.104 + 0.994i)15^-s + (0.669 − 0.743i)16^-s + (0.978 + 0.207i)17^-s + (0.809 + 0.587i)18^-s + (0.104 + 0.994i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$143$    =    $11 \cdot 13$
Sign:	$0.735 + 0.677i$
Analytic conductor:	$15.3674$
Root analytic conductor:	$15.3674$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{143} (74, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 143,\ (1:\ ),\ 0.735 + 0.677i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$4.200477779 + 1.638399186i$
$L(1)$	$\approx$	$2.517075101 + 0.5071180270i$

Euler product

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

	$p$	$F_p(T)$
bad	11	$1$
	13	$1$
good	2	$1 + (0.978 - 0.207i)T$
	3	$1 + (0.913 + 0.406i)T$
	5	$1 + (0.309 + 0.951i)T$
	7	$1 + (-0.913 + 0.406i)T$
	17	$1 + (0.978 + 0.207i)T$
	19	$1 + (0.104 + 0.994i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (0.104 - 0.994i)T$
	31	$1 + (0.309 - 0.951i)T$

Imaginary part of the first few zeros on the critical line

−28.17349925934869804098145498107, −26.51679397837374638357482420721, −25.56705683538424164289454396454, −25.04939824411002486153661786527, −23.93459888413805591505298938826, −23.30599190499828571902864885669, −21.84215501256502652861770393889, −20.99725456403112042628365611477, −19.990985531784482844875390092010, −19.48330336882719784510374289964, −17.74199715737795305812194357930, −16.435526757506027221973993686268, −15.73246649764482017720444598137, −14.39375044163493263603834626716, −13.50436967090618120326260129794, −12.8629133983784491280839791359, −11.95362695679929239443847630440, −10.09667214387707605623594610150, −8.940843470253624121593594363629, −7.6604391974053215836946636076, −6.64379271791021899115405530389, −5.29659347424948465013738078190, −3.91203388184282661122654671926, −2.85245337111281502467113002038, −1.2941244673100983730508148317, 2.096077616471690955366696051627, 3.08392073973293100829331083088, 3.942443669285619909468182310188, 5.65429713383079518562484773246, 6.69719759958708498165714562023, 8.00809062808935874381575007387, 9.85308612912031550725738647349, 10.25696572479450289698857383188, 11.823589666163398143318567349893, 13.018330269315751741036227322427, 13.98492935218703199466467414941, 14.766104138774527596190641799018, 15.59936101138580924600898327513, 16.63841239760023206966955918338, 18.7680912363026199828554480610, 19.14224431623371461425699986200, 20.46875667506887958390335384639, 21.26993246799346011083356181751, 22.29007296848907175037613921538, 22.78717906635545687008796634416, 24.287344962569638790190093444203, 25.465275964661057147510924879877, 25.770382704561328076372772448674, 27.02108720659230123344036271270, 28.37248314830797173373580972096

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