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Properties

Label 1-260-260.67-r1-0-0
Degree $1$
Conductor $260$
Sign $0.868 - 0.496i$
Analytic cond. $27.9408$
Root an. cond. $27.9408$
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 + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s + i·21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − i·31-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s + i·21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − i·31-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ 0.868 - 0.496i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.176588553 - 0.3126283321i\)
\(L(\frac12)\) \(\approx\) \(1.176588553 - 0.3126283321i\)
\(L(1)\) \(\approx\) \(0.8181400540 - 0.1392047768i\)
\(L(1)\) \(\approx\) \(0.8181400540 - 0.1392047768i\)

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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−25.74051514270216838898466501232, −24.762194035697102881953326679309, −23.88189726438720570102661262797, −22.77036675910552276230671383260, −22.08709872803773150390079633666, −21.50532754576557093946118822415, −20.311613415898672923024845578958, −19.1577198900454892553053292803, −18.31764270307869965573979615714, −17.24655630554638215098903463906, −16.49343968636314731869327533139, −15.53492788222630022517097321309, −14.80776695354856646702812089537, −13.33481562688364287564175748551, −12.32491673912847595602946493279, −11.47872498082032906868698877828, −10.62311050923884168797042587054, −9.33331646768923479871236356098, −8.755126542282545346711999002390, −6.845790381206139029149763616723, −6.184841659015656128052189521383, −5.057528605963698931943489875239, −3.96959140224675530470274649620, −2.58275824513833893095831272142, −0.72716522764090575021804377744, 0.71027992925788482619386334832, 1.96911324996443159274308149573, 3.81383552698561334445371343995, 4.79365082525898523600116909679, 6.30007972294808970119858637731, 6.80541834307556519392428390609, 7.95244091476010517625326870950, 9.40450524056959562177922654333, 10.49951634833287401356883095403, 11.28040303455125720797753538927, 12.45328592396924664347316396529, 13.12390810392248584016362612747, 14.1853783551156643073211992282, 15.43873049787986152917397371307, 16.59296843381139077303844807801, 17.21296001585906584334105018434, 17.968704442475110315028147573914, 19.34201640603230169587210189335, 19.71188008038624308543223871736, 21.146166129413205400831175106661, 22.140725692741031678105518750154, 23.065566714680387807967389879134, 23.46752930229727595065229192448, 24.6912412904959173464687376083, 25.3753925074250948543144050901

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