L-function 1-297-297.59-r1-0-0

• Label: 1-297-297.59-r1-0-0
• Degree: $1$
• Conductor: $297$
• Sign: $0.947 + 0.320i$
• Analytic cond.: $31.9170$
• Root an. cond.: $31.9170$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.990 + 0.139i)2^-s + (0.961 − 0.275i)4^-s + (0.374 + 0.927i)5^-s + (0.559 − 0.829i)7^-s + (−0.913 + 0.406i)8^-s + (−0.5 − 0.866i)10^-s + (−0.882 + 0.469i)13^-s + (−0.438 + 0.898i)14^-s + (0.848 − 0.529i)16^-s + (0.978 − 0.207i)17^-s + (0.913 − 0.406i)19^-s + (0.615 + 0.788i)20^-s + (0.939 + 0.342i)23^-s + (−0.719 + 0.694i)25^-s + (0.809 − 0.587i)26^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$297$    =    $3^{3} \cdot 11$
Sign:	$0.947 + 0.320i$
Analytic conductor:	$31.9170$
Root analytic conductor:	$31.9170$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{297} (59, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 297,\ (1:\ ),\ 0.947 + 0.320i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.427747102 + 0.2350242334i$
$L(1)$	$\approx$	$0.8663790816 + 0.1003951290i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	11	$1$
good	2	$1 + (-0.990 + 0.139i)T$
	5	$1 + (0.374 + 0.927i)T$
	7	$1 + (0.559 - 0.829i)T$
	13	$1 + (-0.882 + 0.469i)T$
	17	$1 + (0.978 - 0.207i)T$
	19	$1 + (0.913 - 0.406i)T$
	23	$1 + (0.939 + 0.342i)T$
	29	$1 + (-0.438 - 0.898i)T$
	31	$1 + (0.0348 - 0.999i)T$

Imaginary part of the first few zeros on the critical line

−24.95004482014078224833016754873, −24.74145313343757802391682510872, −23.6827765056190080014372952820, −22.125252530276809545559864475145, −21.24914349534620766261506036672, −20.58998800901287598426988052983, −19.70368034857437604925166572411, −18.67351205764415929341818570936, −17.8616651118933068016466810282, −17.03019185417777040028055261997, −16.26770950260955959998653758149, −15.23613262106528316989111749126, −14.25058223113465044733077136302, −12.52381884934266066896253037036, −12.26573687018402041468264019896, −10.9998516121390051360606120537, −9.861955863447414737971888762846, −9.09792747611711601313339545123, −8.22982843381031781545245562082, −7.327491784740044436448041671693, −5.78529046573240277703261547202, −5.03335301415157015603311963060, −3.15039093072990561823059669617, −1.9202271940817942592309001150, −0.86684974901412937950062052120, 0.86424259226114871191539004045, 2.17168733664419236670614214004, 3.32101107052146765946162558127, 5.04977700466073844733503392340, 6.34513142058193420137040964232, 7.34889943024451931106911185524, 7.84688422557679258374144003495, 9.54033752584710696760483199088, 9.94753227866798539981655740417, 11.16693592392809959890010793671, 11.661110349397004408805762390624, 13.37767911188879116956256305633, 14.4962961854211564074042998272, 15.00943129582731967754482105753, 16.40774019590358974411246938157, 17.16810460924600136411025485189, 17.90688280169292671568200058693, 18.8185125233558334412273993083, 19.56988333631610453009066689011, 20.65317709645867338684309715202, 21.39221835745471038620856964410, 22.587092561055180662659196522449, 23.61800594729866253984201896828, 24.51030196761804191646128761713, 25.39830006220394719527772102445

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