L-function 1-1183-1183.9-r0-0-0

• Label: 1-1183-1183.9-r0-0-0
• Degree: $1$
• Conductor: $1183$
• Sign: $0.109 + 0.993i$
• Analytic cond.: $5.49382$
• Root an. cond.: $5.49382$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.0402 + 0.999i)2^-s + (−0.970 + 0.239i)3^-s + (−0.996 − 0.0804i)4^-s + (0.987 − 0.160i)5^-s + (−0.200 − 0.979i)6^-s + (0.120 − 0.992i)8^-s + (0.885 − 0.464i)9^-s + (0.120 + 0.992i)10^-s + (0.885 + 0.464i)11^-s + (0.987 − 0.160i)12^-s + (−0.919 + 0.391i)15^-s + (0.987 + 0.160i)16^-s + (−0.919 + 0.391i)17^-s + (0.428 + 0.903i)18^-s + 19^-s + (−0.996 + 0.0804i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1183$    =    $7 \cdot 13^{2}$
Sign:	$0.109 + 0.993i$
Analytic conductor:	$5.49382$
Root analytic conductor:	$5.49382$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1183} (9, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1183,\ (0:\ ),\ 0.109 + 0.993i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9096637536 + 0.8146240979i$
$L(1)$	$\approx$	$0.7885733151 + 0.4581847538i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	13	$1$
good	2	$1 + (-0.0402 + 0.999i)T$
	3	$1 + (-0.970 + 0.239i)T$
	5	$1 + (0.987 - 0.160i)T$
	11	$1 + (0.885 + 0.464i)T$
	17	$1 + (-0.919 + 0.391i)T$
	19	$1 + T$
	23	$1 + (-0.5 + 0.866i)T$
	29	$1 + (-0.845 - 0.534i)T$
	31	$1 + (-0.200 - 0.979i)T$

Imaginary part of the first few zeros on the critical line

−21.11297305864645222031555641431, −20.36272379391398963402053312464, −19.47648726314811984930084104963, −18.55796614106620539224332577316, −18.07600364351060299967019027889, −17.408541351989197645342855706062, −16.754619547899513627245908641913, −15.86679151828263888848498644883, −14.374370373303330286129378695911, −13.92162449008729685617028115357, −13.03344736615326792346370411556, −12.39317861470774420011199310279, −11.48808245674648932650142520327, −10.95216726807884779515303038568, −10.14321348018336513495825748933, −9.38754422681368082530196260499, −8.64792661027376975581148838637, −7.266736220898946028926980441625, −6.38068943315635979303569086700, −5.55153414681795712313402769712, −4.80426360722020317471928127244, −3.79024518445740832929649037657, −2.60749898376575735058405017572, −1.67411370598185141186809175411, −0.830917114658708659125333079728, 0.88789108110360893086521261274, 1.98002748354871748160219238269, 3.8442467159889073019394997981, 4.45921896907264364889956823412, 5.67105264219988800341754524995, 5.79779427017528029920626609318, 6.8615939335222564371331573199, 7.474730131213433035496055897, 8.89303000560284444136639103608, 9.50653681975053893824562283706, 10.062635383019679975498233744622, 11.15734612744287673608776641761, 12.09841484066802818902071372363, 13.02275145659747483147472676562, 13.61394893429707113192405163437, 14.58018632133914775082475095420, 15.317755282353305308214635282693, 16.16580897038838746992760283505, 16.8292573120179841985719577385, 17.576546498107868322480682686419, 17.81152318287144372758615164611, 18.70816038162359259598172851658, 19.81454901578799546379414008505, 20.86901501842005846813312370599, 21.807917170990325841562462508517

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