L-function 1-260-260.219-r0-0-0

• Label: 1-260-260.219-r0-0-0
• Degree: $1$
• Conductor: $260$
• Sign: $-0.466 - 0.884i$
• Analytic cond.: $1.20743$
• Root an. cond.: $1.20743$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.5 + 0.866i)3^-s + (−0.866 + 0.5i)7^-s + (−0.5 − 0.866i)9^-s + (−0.866 − 0.5i)11^-s + (−0.5 − 0.866i)17^-s + (−0.866 + 0.5i)19^-s − i·21^-s + (0.5 − 0.866i)23^-s + 27^-s + (−0.5 + 0.866i)29^-s − i·31^-s + (0.866 − 0.5i)33^-s + (−0.866 − 0.5i)37^-s + (−0.866 − 0.5i)41^-s + (0.5 + 0.866i)43^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$260$    =    $2^{2} \cdot 5 \cdot 13$
Sign:	$-0.466 - 0.884i$
Analytic conductor:	$1.20743$
Root analytic conductor:	$1.20743$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{260} (219, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 260,\ (0:\ ),\ -0.466 - 0.884i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.07960496739 - 0.1319303693i$
$L(1)$	$\approx$	$0.5691113059 + 0.09804384990i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	13	$1$
good	3	$1 + (-0.5 + 0.866i)T$
	7	$1 + (-0.866 + 0.5i)T$
	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

−25.87384558202553337550336599149, −25.46248652983291924847545405428, −24.100035175414034263087064579495, −23.51285156404405833963041914067, −22.754932335798813698767079177268, −21.811589784154823836181491169364, −20.581101877652448890041476192690, −19.44679153639445324228445715052, −19.00600650177049555996906506011, −17.68928148362796963898589564762, −17.18316686725067969210165300977, −16.04381289690027853631621135088, −15.07129719260820927851359979762, −13.54298764888125064928633340285, −13.0836561026300271569279011065, −12.197213306692703620028950637454, −10.93604000730196377837836894782, −10.19046717722065489638443614659, −8.76460928935189006071499609208, −7.562347640728976261630758360020, −6.76340252065107718334852347715, −5.79394797871571693578748968625, −4.52394518210055813828168969324, −2.983601803451774352067748785412, −1.682683313410262093915212175358, 0.10687052407198912602506435249, 2.55426410527838404516915340327, 3.6211068822902968404704388328, 4.92126635656320587641126068241, 5.839986690326958168353329160174, 6.840300134599560837357917055681, 8.479196533899244241779588382154, 9.34750525599006963371791236829, 10.35665898517963985148188222375, 11.158329084798259220996863394235, 12.32008950283873085076666210756, 13.20683645993651302497313810378, 14.5660347206380834881171589274, 15.5520714239741575352247095009, 16.19976083337000478137909158358, 17.00577460507199556941753567101, 18.26239684911887171506760633885, 19.0035746799212549341798883222, 20.32857311652906560501253144339, 21.08070931530918711598039223900, 22.03587539711765308081826981801, 22.69971499242905284220972367722, 23.55772006214636122808727004951, 24.71152386490743251430196499714, 25.83254112138581465227873112478

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