L-function 1-3e3-27.14-r1-0-0

• Label: 1-3e3-27.14-r1-0-0
• Degree: $1$
• Conductor: $27$
• Sign: $0.686 - 0.727i$
• Analytic cond.: $2.90155$
• Root an. cond.: $2.90155$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.939 − 0.342i)2^-s + (0.766 − 0.642i)4^-s + (−0.173 − 0.984i)5^-s + (0.766 + 0.642i)7^-s + (0.5 − 0.866i)8^-s + (−0.5 − 0.866i)10^-s + (−0.173 + 0.984i)11^-s + (−0.939 − 0.342i)13^-s + (0.939 + 0.342i)14^-s + (0.173 − 0.984i)16^-s + (0.5 + 0.866i)17^-s + (−0.5 + 0.866i)19^-s + (−0.766 − 0.642i)20^-s + (0.173 + 0.984i)22^-s + (−0.766 + 0.642i)23^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$27$    =    $3^{3}$
Sign:	$0.686 - 0.727i$
Analytic conductor:	$2.90155$
Root analytic conductor:	$2.90155$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{27} (14, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 27,\ (1:\ ),\ 0.686 - 0.727i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.071104870 - 0.8933874351i$
$L(1)$	$\approx$	$1.693468257 - 0.5069911147i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
good	2	$1 + (0.939 - 0.342i)T$
	5	$1 + (-0.173 - 0.984i)T$
	7	$1 + (0.766 + 0.642i)T$
	11	$1 + (-0.173 + 0.984i)T$
	13	$1 + (-0.939 - 0.342i)T$
	17	$1 + (0.5 + 0.866i)T$
	19	$1 + (-0.5 + 0.866i)T$
	23	$1 + (-0.766 + 0.642i)T$
	29	$1 + (0.939 - 0.342i)T$

Imaginary part of the first few zeros on the critical line

−37.95795327445094493223772281054, −36.44802792303360602147236236885, −34.42007554618066471941828844409, −34.15019652323261997852977506671, −32.64280915195817769583370829302, −31.36834947262606727605218485366, −30.20374539284429544169813633525, −29.4084404225114357787276165107, −27.08801668705320388410210023406, −26.14393830784178631658583858488, −24.490886264181400886607138246431, −23.48146292953663819578070443289, −22.1926824742599547555411765847, −21.10207807132164329616702587115, −19.474105673837352034479987755721, −17.65233965292251957060418623829, −16.13394544126673928874690926553, −14.58147314594082583881212134431, −13.82344868704737974178134617838, −11.83148258646725015257941183458, −10.664298392187542185048700515350, −7.89316727903529802303258863456, −6.60501479198176008537476565692, −4.67511275894351686242751712383, −2.860516751384937779069980089754, 1.92612712073415051907750420112, 4.36520335611636501165930356609, 5.601706788614045471861445311395, 7.91438509969654085095805260558, 9.991470414909572147048294478793, 11.9355147367484747346440851740, 12.68007819388464012863046390259, 14.50437732966114659069477724520, 15.63388976653377013031231855848, 17.3680276979290959787984954896, 19.378554976055638172940211534073, 20.62589196493439134091083736182, 21.58064217699438680979435816553, 23.16814170117636041896505813731, 24.320587100993145732504595963955, 25.271109151244888044954697463849, 27.67340199926699433354801911443, 28.46321836907213471252069700940, 29.95543595323483945612623839767, 31.26567410102111338560968710387, 32.03322842928821457711962847547, 33.39356908290097460807293025414, 34.564308501135083620274381586638, 36.31841020614197660949147835154, 37.5133816611696645329604491317

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