Properties

Label 1-260-260.119-r0-0-0
Degree $1$
Conductor $260$
Sign $0.878 + 0.477i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.878 + 0.477i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ 0.878 + 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.133809563 + 0.2881862283i\)
\(L(\frac12)\) \(\approx\) \(1.133809563 + 0.2881862283i\)
\(L(1)\) \(\approx\) \(1.017894033 + 0.1908600893i\)
\(L(1)\) \(\approx\) \(1.017894033 + 0.1908600893i\)

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

−25.575887400125658522334844664026, −24.55355544691261862891799072385, −24.29757649711228820703612604041, −23.14758192805978071687003621444, −22.21304108201135396965693390645, −21.426840104404011181652184097383, −20.177251223236626209034584452550, −19.18044503656860197660773189330, −18.45158474630218728483683067476, −17.446963671115022213784603980844, −16.885905812750247302647485255430, −15.529466945393176225686229693788, −14.433390146393828867011823366677, −13.57734245079875785451539794154, −12.4852982195627547239206599332, −11.54608989751826464291981449158, −11.013590536196942322115599596272, −9.37891621274718231607618851320, −8.267168826395085642414729895283, −7.41951939840336958792112657884, −6.13365743408468304308273429251, −5.42882485802680428360898666433, −3.97138509390705644730172655838, −2.27746367805410750682095602825, −1.20987697307964358046480869072, 1.17970168376196506091433908364, 3.03520786638820151509503468266, 4.43739308262826677042524546261, 4.935561297843675742995524602181, 6.39285042745160355459866739567, 7.44870735137700935803983412291, 8.906725649780829042452457216193, 9.66677096299150345987905142334, 10.92373206751170123500564575110, 11.43633917515500220996192862785, 12.568078659124740596316852635014, 14.09177034395579959712967953021, 14.69196528929596766471588540134, 15.804662596432902290169912018624, 16.69070830062179336285180696769, 17.56417585296029427435293989373, 18.242948490373739001781042373565, 19.94103035202902786221281245345, 20.43465821524717336053013385348, 21.431920779279573032764830213112, 22.35894426457422443066526915135, 23.02699060254615349950523398400, 24.10140218782164527008110349259, 24.977175530176464487531575221285, 26.23252844671890493879277993053

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