L-function 1-143-143.129-r1-0-0

• Label: 1-143-143.129-r1-0-0
• Degree: $1$
• Conductor: $143$
• Sign: $0.530 + 0.847i$
• 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

+ (0.309 − 0.951i)2^-s + (−0.809 + 0.587i)3^-s + (−0.809 − 0.587i)4^-s + (−0.309 − 0.951i)5^-s + (0.309 + 0.951i)6^-s + (−0.809 − 0.587i)7^-s + (−0.809 + 0.587i)8^-s + (0.309 − 0.951i)9^-s − 10^-s + 12^-s + (−0.809 + 0.587i)14^-s + (0.809 + 0.587i)15^-s + (0.309 + 0.951i)16^-s + (−0.309 − 0.951i)17^-s + (−0.809 − 0.587i)18^-s + (−0.809 + 0.587i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$143$    =    $11 \cdot 13$
Sign:	$0.530 + 0.847i$
Analytic conductor:	$15.3674$
Root analytic conductor:	$15.3674$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{143} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 143,\ (1:\ ),\ 0.530 + 0.847i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1306830444 + 0.07241491096i$
$L(1)$	$\approx$	$0.5237271712 - 0.3274647246i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−27.828492641589487760955978901727, −26.72795186314646307759324441607, −25.72634595924408506982372308637, −24.960263105395905931379276370755, −23.723279946660407199746327552181, −23.13219984400068934097517624054, −22.179865932771270880761337449378, −21.65442128226436022990957941934, −19.32371454754670997414703215844, −18.73982021425492382075586739512, −17.694177896349616896399510953155, −16.80117787882228889515337923533, −15.62642769350255460597827793745, −14.91138969318552023439880305488, −13.452308703834402920300134382996, −12.67653967601210367771442117105, −11.54271755164392064680586214208, −10.25405639936239077136833356139, −8.68284756282232328026783077793, −7.36758127948836475060794799855, −6.50371291602102460787548891705, −5.80076757216193416826483519936, −4.2364227758913307857773322498, −2.668892565654190650330968863821, −0.069266447936949241847383257418, 1.08974519970853279256000106276, 3.28836518476046724293561602272, 4.399469797144225069514464767592, 5.23652248710850141217067638605, 6.661157134159330587167645655, 8.719045930616089289606247006197, 9.70756786774637167839529791935, 10.62725655636466641968456050888, 11.74429191134332930166633611527, 12.604452600402011484759664144, 13.48498496538353704860501769874, 15.035352597544263428501620218291, 16.24040977988834628165935118863, 17.00469354882354760034909716111, 18.24145580805315622297628037458, 19.52696724204820574318101988015, 20.37531831084231045874727070359, 21.203545441213311147727726626703, 22.22331392192481730806750634386, 23.260372583816819467418026915023, 23.61196873071069017764837073953, 25.18795042420261010172104260395, 26.94344157914545745304905945472, 27.26768120899851875982255795862, 28.52322020346310148050202709775

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