L-function 1-3311-3311.236-r1-0-0

• Label: 1-3311-3311.236-r1-0-0
• Degree: $1$
• Conductor: $3311$
• Sign: $0.843 - 0.537i$
• Analytic cond.: $355.816$
• Root an. cond.: $355.816$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.251 + 0.967i)2^-s + (0.772 − 0.635i)3^-s + (−0.873 + 0.486i)4^-s + (0.337 + 0.941i)5^-s + (0.809 + 0.587i)6^-s + (−0.691 − 0.722i)8^-s + (0.193 − 0.981i)9^-s + (−0.826 + 0.563i)10^-s + (−0.365 + 0.930i)12^-s + (−0.473 + 0.880i)13^-s + (0.858 + 0.512i)15^-s + (0.525 − 0.850i)16^-s + (0.999 + 0.0299i)17^-s + (0.998 − 0.0598i)18^-s + (−0.575 − 0.817i)19^-s + (−0.753 − 0.657i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3311$    =    $7 \cdot 11 \cdot 43$
Sign:	$0.843 - 0.537i$
Analytic conductor:	$355.816$
Root analytic conductor:	$355.816$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3311} (236, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3311,\ (1:\ ),\ 0.843 - 0.537i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.594452373 - 0.4650521613i$
$L(1)$	$\approx$	$1.191181291 + 0.5129496432i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	11	$1$
	43	$1$
good	2	$1 + (0.251 + 0.967i)T$
	3	$1 + (0.772 - 0.635i)T$
	5	$1 + (0.337 + 0.941i)T$
	13	$1 + (-0.473 + 0.880i)T$
	17	$1 + (0.999 + 0.0299i)T$
	19	$1 + (-0.575 - 0.817i)T$
	23	$1 + (-0.733 + 0.680i)T$
	29	$1 + (0.936 - 0.351i)T$
	31	$1 + (-0.712 + 0.701i)T$

Imaginary part of the first few zeros on the critical line

−19.10326527823085026089356281860, −18.168665941624318339459815330833, −17.45670558926073320812368329542, −16.58772347656235234240894136070, −16.05388419043969918911936738326, −14.99932667706192200119165038812, −14.46542331653009008532634580784, −13.89969139982336458238438679377, −13.022753964303497109694127597846, −12.56005243915404524720275705476, −11.94524056175605995538914486792, −10.78838410197392793422089197897, −10.209365729881900343452087253611, −9.70674111788723613105938967928, −9.049129046486616628259791933732, −8.13374070874752243895330980592, −7.93290836690090161005713232915, −6.22311702103345459687942816303, −5.36401115774154588222364520451, −4.8697471943657045491779910870, −4.04565969440609519046764232167, −3.38762861497733125655899611854, −2.497275718523862162819078623123, −1.795172686971596265471747176588, −0.8712629731722240379122058077, 0.220811233789042704464875351811, 1.56928375375129838738316846210, 2.45478159335700647395775434122, 3.2960974419748920931971098499, 3.91735944222722709332227395267, 4.96392260615163841207371429507, 5.924382449813862878544474970137, 6.59920210512252227520459216961, 7.136858943659037231677071531017, 7.71237678316065756100430906610, 8.52101657493049891317680668659, 9.31879957064946310592093015080, 9.829322465900139517609498082885, 10.81941096961529608426129717941, 12.076895476111038962959902482, 12.32847070736497139546863534282, 13.56122167283422847254898803462, 13.81569230165615662624789009311, 14.40538215339706543187448867398, 15.05335436898469209944078050646, 15.59810719655520950563643107713, 16.55264767155735613039074859027, 17.371965319635421232880798021186, 17.90509548719107138257940887272, 18.52254402755988366669494011334

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