L-function 1-273-273.20-r1-0-0

• Label: 1-273-273.20-r1-0-0
• Degree: $1$
• Conductor: $273$
• Sign: $0.852 + 0.522i$
• Analytic cond.: $29.3379$
• Root an. cond.: $29.3379$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 + 0.5i)2^-s + (0.5 − 0.866i)4^-s + i·5^-s + i·8^-s + (−0.5 − 0.866i)10^-s + (0.866 − 0.5i)11^-s + (−0.5 − 0.866i)16^-s + (0.5 − 0.866i)17^-s + (0.866 + 0.5i)19^-s + (0.866 + 0.5i)20^-s + (−0.5 + 0.866i)22^-s + (−0.5 − 0.866i)23^-s − 25^-s + (0.5 + 0.866i)29^-s − i·31^-s + (0.866 + 0.5i)32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$273$    =    $3 \cdot 7 \cdot 13$
Sign:	$0.852 + 0.522i$
Analytic conductor:	$29.3379$
Root analytic conductor:	$29.3379$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{273} (20, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 273,\ (1:\ ),\ 0.852 + 0.522i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.282477415 + 0.3614261739i$
$L(1)$	$\approx$	$0.8063256420 + 0.2110008989i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	13	$1$
good	2	$1 + (-0.866 + 0.5i)T$
	5	$1 + iT$
	11	$1 + (0.866 - 0.5i)T$
	17	$1 + (0.5 - 0.866i)T$
	19	$1 + (0.866 + 0.5i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (0.5 + 0.866i)T$
	31	$1 - iT$
	37	$1 + (-0.866 + 0.5i)T$

Imaginary part of the first few zeros on the critical line

−25.41803444811034925477286559377, −24.75983348201019502266366187587, −23.792788969117158197332147249122, −22.487590898406096652223330674, −21.44897131170896309483586040833, −20.74767001911591486388480358220, −19.682473829429285852466240898942, −19.38122411721874237328290090466, −17.76383838134717468929030966432, −17.40091358136894862809193525766, −16.32139385228150077756357440737, −15.59315619925057909465707888163, −14.10562163935242409421186383679, −12.85282393088727691653259175220, −12.14734991437298243901703969433, −11.29577469593156094030429494644, −9.929728201544637887423529519314, −9.29305419156054460481871067801, −8.30374270161896367396630965668, −7.3729266446944454065249759380, −6.03526133764313619827392525321, −4.53906975251464164187780029132, −3.44075990981077973770726093426, −1.827543024427916655243258651028, −0.90502645176782729644276870263, 0.77424037566086484264407401365, 2.28925049257603702923624357356, 3.58545886978054068359630709594, 5.36580425176134717454739025034, 6.419394053535866640866672670, 7.198027690093791811022028858324, 8.2288749239726912319432774874, 9.39286967242287362562629378780, 10.226199506116894351587094173720, 11.21270939051572538178767066871, 12.046481005810714868583914420507, 14.05464164060995936581741980357, 14.30461642801769108008188894526, 15.532187818459373535099388305011, 16.37052381121447978504622362185, 17.320527865267943100426604244354, 18.36968362984864950080406815246, 18.829569484081481090089768965270, 19.85183691657615182510881413302, 20.79288784759348183493613797329, 22.19638258938501537714058099450, 22.817905279271053756259842715695, 23.9898058706500565281226121952, 24.84049240192059502839831334744, 25.64965942071834720031067766925

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