Properties

Label 1-260-260.243-r0-0-0
Degree $1$
Conductor $260$
Sign $-0.223 + 0.974i$
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  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + 21-s + (−0.866 + 0.5i)23-s i·27-s + (0.5 + 0.866i)29-s − 31-s + (−0.866 − 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + 21-s + (−0.866 + 0.5i)23-s i·27-s + (0.5 + 0.866i)29-s − 31-s + (−0.866 − 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ -0.223 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3988082301 + 0.5006093908i\)
\(L(\frac12)\) \(\approx\) \(0.3988082301 + 0.5006093908i\)
\(L(1)\) \(\approx\) \(0.6741564420 + 0.2138328458i\)
\(L(1)\) \(\approx\) \(0.6741564420 + 0.2138328458i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−25.446579158762309514897839991185, −24.66582172444303836165100990859, −23.7803704347107939256941164749, −22.86172688955641529069193050771, −22.0859866669198566668326947338, −21.38943274194871323647899537202, −19.84662110542608496693304328279, −19.01991195408898644770116777424, −18.361869735597485526903870788263, −17.21379384053722226894855362752, −16.39473078232288148461806562143, −15.672613937281565510930004565113, −14.204274989743163056848173486646, −13.227017534906449253592354717880, −12.30400154679512119989847502140, −11.550586029229247829410824819475, −10.46805889467902236031450926467, −9.36620240438535130757828547394, −8.16561234255406820432803591166, −6.8499222105825376533283083245, −6.114262451197214070831759235787, −5.15577212574427143942693712694, −3.63172080360529662358804315261, −2.22599866306494762864171271818, −0.51927491422834612087390491712, 1.456104556506948437692260389022, 3.472399827383373910081998457730, 4.27951356995002056899582063569, 5.62273554055950075286347462496, 6.51706500039352528182761669834, 7.53781910243128907595996192675, 9.18186625505281297261505719356, 10.06980437474521783918337333104, 10.71016257349533859796715711151, 12.18630717506620529293396462998, 12.548520286745757956720097891222, 14.044536523447736379328654185600, 15.087728546701573480085182394480, 16.10793101418270935834756113479, 16.829967494552336711923181090518, 17.60100964957706480406501156126, 18.72028558240502369194485960724, 19.79400497218460681194521186828, 20.71648726421664347707595635111, 21.75215957527745711291445446859, 22.59625765982959186052393739295, 23.23353086151461142205090616252, 24.04824319623686287530525221854, 25.54638816432290203135091633362, 26.005055659278828501338985910764

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