Properties

Label 1-260-260.19-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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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: $\chi_{260} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ -0.466 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07960496739 + 0.1319303693i\)
\(L(\frac12)\) \(\approx\) \(0.07960496739 + 0.1319303693i\)
\(L(1)\) \(\approx\) \(0.5691113059 - 0.09804384990i\)
\(L(1)\) \(\approx\) \(0.5691113059 - 0.09804384990i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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