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Properties

Label 1-260-260.23-r0-0-0
Degree $1$
Conductor $260$
Sign $0.0427 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.0427 + 0.999i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ 0.0427 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9860420883 + 0.9447182246i\)
\(L(\frac12)\) \(\approx\) \(0.9860420883 + 0.9447182246i\)
\(L(1)\) \(\approx\) \(1.131922494 + 0.4567084276i\)
\(L(1)\) \(\approx\) \(1.131922494 + 0.4567084276i\)

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

−25.759385532697043698242072318, −24.72805025126575278497902131530, −24.02437609673621802065410611428, −23.04639674275366295174068643552, −21.99170264880167663043162341733, −20.90356407223538421572793054691, −20.028810163387872038562010505, −19.30132071516846944166398965888, −18.49316694106154682166387332139, −17.45442255353748170720045606951, −16.158149301401741785684043181395, −15.47606829076729186955865284005, −14.17226202914311556805734992709, −13.417985456131657114565463446811, −12.8108433180492198791130557213, −11.44932787422438738950575856861, −10.25916028502551930612986429486, −9.18200970702995926660300021451, −8.36142991853384716910322912065, −7.1136428879527597852014311409, −6.45135527209661992321220555615, −4.79821228872998484391258152151, −3.34442058020177381264717211907, −2.655659081368776140051856566505, −0.86836629113540737847735368926, 2.00925051803290075693287298602, 3.03138925251063456632497483079, 4.14872905523266827390138890842, 5.34476343417247707631673028977, 6.73033870555618602871924112171, 7.86973927152153125978835118471, 8.92411766646962275593024972335, 9.77901744083760965079433509499, 10.560735098936221457750660392944, 12.08843948487443193764932647102, 13.05365667304950656468401433760, 13.88174579460360323743336436798, 15.2460850794441464273976676706, 15.5047506277510826734009428594, 16.64096728423182100396880114946, 17.88015888379420334674346507442, 19.01170953763333568706497887430, 19.64575547945102531104404256136, 20.65292026768975466696376769302, 21.4086878882167373651931752218, 22.42942647683888702816997982764, 23.18196703211816819784658091646, 24.68810748202938454899496755826, 25.209100408339075277437067915316, 26.19205237625802405015941971100

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