L-function 1-1455-1455.158-r0-0-0

• Label: 1-1455-1455.158-r0-0-0
• Degree: $1$
• Conductor: $1455$
• Sign: $-0.835 - 0.550i$
• Analytic cond.: $6.75699$
• Root an. cond.: $6.75699$
• 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 + (−0.866 + 0.5i)7^-s − i·8^-s + (0.5 − 0.866i)11^-s + (0.866 − 0.5i)13^-s + (−0.5 + 0.866i)14^-s + (−0.5 − 0.866i)16^-s + (−0.866 − 0.5i)17^-s − 19^-s − i·22^-s + (0.866 + 0.5i)23^-s + (0.5 − 0.866i)26^-s + i·28^-s + (−0.5 − 0.866i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1455$    =    $3 \cdot 5 \cdot 97$
Sign:	$-0.835 - 0.550i$
Analytic conductor:	$6.75699$
Root analytic conductor:	$6.75699$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1455} (158, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1455,\ (0:\ ),\ -0.835 - 0.550i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5380905472 - 1.794631738i$
$L(1)$	$\approx$	$1.258073037 - 0.7632706124i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
	97	$1$
good	2	$1 + (0.866 - 0.5i)T$
	7	$1 + (-0.866 + 0.5i)T$
	11	$1 + (0.5 - 0.866i)T$
	13	$1 + (0.866 - 0.5i)T$
	17	$1 + (-0.866 - 0.5i)T$
	19	$1 - T$
	23	$1 + (0.866 + 0.5i)T$
	29	$1 + (-0.5 - 0.866i)T$
	31	$1 + (-0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

−21.15118008028707649115690877812, −20.320636121241749524480890699371, −19.71710102864112248280373713085, −18.85283471239855547913223176696, −17.6778971592092250409252533878, −17.12910481529445568244612436648, −16.31820081981094542820392484791, −15.749491273900220135873081648468, −14.86445159174952450853818914643, −14.27018309296754812244534067603, −13.30347600763356347385753301595, −12.81241085707563821476361376772, −12.17161087693306573480186509374, −10.98881379646705375036718981305, −10.542162232710008643098845421633, −9.10379532825460957665212156948, −8.69265191940077030782902598395, −7.34272793219696082361611993400, −6.77007041668284652591850978917, −6.25777560041449916504737916329, −5.14554592590949956684364459505, −4.11207757655898962912372560905, −3.769136087323160002990982592443, −2.579160739411999704304299962787, −1.56743030889104601687067875313, 0.48977312717600109196193159733, 1.76999353103701326850076762639, 2.80727498653500068316007453639, 3.46694564607871680979365731263, 4.29476037928873399318971492160, 5.38244301368508398438399220539, 6.22221004547505511475018056961, 6.55172116537211606724889102114, 7.887498517842419239841765346475, 9.10289068658851419329495987967, 9.468473695037928209850124644049, 10.847083931336196345514260889712, 11.07155402957749084446080920579, 12.075027328441040265912487481103, 12.92521471388295582673186734176, 13.3786296551688509903018575954, 14.12253910888629406183264844038, 15.25170522241723952666929288712, 15.557785222015823929287237048774, 16.426494104466886314848711620373, 17.31723734969849080671804023915, 18.565398339123499317304991161591, 19.004349019984289904474714541301, 19.67114155130038157239792114304, 20.51804890640958928487221213259

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