L-function 1-1480-1480.27-r0-0-0

• Label: 1-1480-1480.27-r0-0-0
• Degree: $1$
• Conductor: $1480$
• Sign: $0.261 + 0.965i$
• Analytic cond.: $6.87309$
• Root an. cond.: $6.87309$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 + 0.5i)3^-s + (0.866 + 0.5i)7^-s + (0.5 + 0.866i)9^-s + 11^-s + (0.866 + 0.5i)13^-s + (−0.866 + 0.5i)17^-s + (−0.5 + 0.866i)19^-s + (0.5 + 0.866i)21^-s − i·23^-s + i·27^-s − 29^-s + 31^-s + (0.866 + 0.5i)33^-s + (0.5 + 0.866i)39^-s + (−0.5 + 0.866i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1480$    =    $2^{3} \cdot 5 \cdot 37$
Sign:	$0.261 + 0.965i$
Analytic conductor:	$6.87309$
Root analytic conductor:	$6.87309$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1480} (27, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1480,\ (0:\ ),\ 0.261 + 0.965i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.074296294 + 1.587664196i$
$L(1)$	$\approx$	$1.560140597 + 0.5479653961i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−20.35712226953831912957126569246, −19.88528968313039133205469653382, −19.09572268238667051285096383458, −18.2594088248092143712669228724, −17.542814267654468884524480496922, −17.043297322213512687175339469720, −15.63494872942117979512582320520, −15.249454889782546505527708910107, −14.31265970416165260448396718637, −13.63886035027610008288997569379, −13.23584745963511944762055479798, −12.08849628761737763881307615611, −11.35775652476355746904280410954, −10.61819639646472963801461940067, −9.42765608658270902722637008093, −8.83133226979306875041186710678, −8.128343512414042149428081792043, −7.22775767544435726145915795240, −6.668175719722171672557294790204, −5.55670034398998495665965233523, −4.328915023182830113214482052459, −3.78452080064938332153321202529, −2.6832275917916785062706246448, −1.70233694643334073589868772920, −0.928040556886346026731283338981, 1.53723684208592308492186726529, 2.04649404605046382904372779818, 3.25025136844035301372177342287, 4.18963664040104006750495487647, 4.62912647028046475037575250980, 5.93379734827555269071192284832, 6.68493586396909130467599986321, 7.89615479460669067712112548663, 8.57798413383891908164535484943, 8.97606320414657687759876005266, 9.97401900096236369855499952070, 10.87082745597060608256055351721, 11.51509250718303814107960034053, 12.462797183757050814377598647118, 13.448159281307343687415458019379, 14.10444173470299571841473832445, 14.94784111228494836488681458056, 15.16048079650482175599708826689, 16.38005742354501361404149249105, 16.83967363913511006444244649505, 17.976891963632462247230056142475, 18.641821304021582548661475825540, 19.389300538161549025680249555382, 20.13416915356659572671615522752, 20.91843716954783717179260106537

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