L-function 1-143-143.137-r1-0-0

• Label: 1-143-143.137-r1-0-0
• Degree: $1$
• Conductor: $143$
• Sign: $0.289 - 0.957i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$143$    =    $11 \cdot 13$
Sign:	$0.289 - 0.957i$
Analytic conductor:	$15.3674$
Root analytic conductor:	$15.3674$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{143} (137, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 143,\ (1:\ ),\ 0.289 - 0.957i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.489034510 - 1.104729436i$
$L(1)$	$\approx$	$1.173620899 - 0.2456673188i$

Euler product

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

	$p$	$F_p(T)$
bad	11	$1$
	13	$1$
good	2	$1 + (-0.406 + 0.913i)T$
	3	$1 + (0.669 - 0.743i)T$
	5	$1 + (0.587 - 0.809i)T$
	7	$1 + (0.743 - 0.669i)T$
	17	$1 + (-0.913 + 0.406i)T$
	19	$1 + (0.207 - 0.978i)T$
	23	$1 + (0.5 + 0.866i)T$
	29	$1 + (-0.978 + 0.207i)T$
	31	$1 + (-0.587 - 0.809i)T$

Imaginary part of the first few zeros on the critical line

−28.207081684834503290133017795287, −27.126239858329682414080513934080, −26.62787359110464049941131210193, −25.56194117979493559837814584550, −24.72302937198305687248611341284, −22.68747340823137910415704634053, −22.04680665777292596035209027687, −21.12093102097760898453559357181, −20.53511834170593621889332455181, −19.23258188518076373861842017771, −18.39501379714107060509592435923, −17.4957373156319082789910683493, −16.13878506220384324766002844870, −14.72487193783433901016421658302, −14.09811035334036084227680308517, −12.82052172000787654431146016818, −11.29388385342197252110940711154, −10.630249185421278319900492822281, −9.48912044936930932850712512494, −8.70057029142545193663043929914, −7.466379371687210581306178139024, −5.44065312227508532764078190010, −4.1002109482896685641984148300, −2.76770860743130764595770617898, −1.9399938416140803167806142451, 0.77677006441035665573504506812, 1.9042544415229415416365353689, 4.21157194159212596612792036257, 5.48513477291720974916444813923, 6.817295487955588273646006767409, 7.78687153521270370685617248139, 8.77913714541632613393761546687, 9.57885366883675462445643628215, 11.17571873634948670145534721984, 13.02458525778752509540778007563, 13.543482961689089755828022706666, 14.56201547734923097405938218968, 15.60956253287708138877851408871, 17.135368524263705687466042228838, 17.516215165156846003102158585921, 18.61322625663265377906519386085, 19.85654013633598205626167146678, 20.51776906766189015221427177460, 21.95079676465302667192245071113, 23.62263859998771767256948811910, 24.0618899337415540391464802183, 24.84662561356218945518482257077, 25.78876879430436636045722010142, 26.57413190380178907884967910472, 27.705702196727918967554860027895

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