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Properties

Label 1-261-261.16-r0-0-0
Degree $1$
Conductor $261$
Sign $0.902 + 0.430i$
Analytic cond. $1.21207$
Root an. cond. $1.21207$
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  + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)7-s + (−0.900 + 0.433i)8-s + (−0.222 + 0.974i)10-s + (0.0747 + 0.997i)11-s + (0.826 − 0.563i)13-s + (0.365 + 0.930i)14-s + (0.0747 + 0.997i)16-s + 17-s + (−0.222 + 0.974i)19-s + (0.826 + 0.563i)20-s + (0.955 + 0.294i)22-s + (0.365 + 0.930i)23-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)7-s + (−0.900 + 0.433i)8-s + (−0.222 + 0.974i)10-s + (0.0747 + 0.997i)11-s + (0.826 − 0.563i)13-s + (0.365 + 0.930i)14-s + (0.0747 + 0.997i)16-s + 17-s + (−0.222 + 0.974i)19-s + (0.826 + 0.563i)20-s + (0.955 + 0.294i)22-s + (0.365 + 0.930i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(1.21207\)
Root analytic conductor: \(1.21207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (0:\ ),\ 0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7866426968 + 0.1778141233i\)
\(L(\frac12)\) \(\approx\) \(0.7866426968 + 0.1778141233i\)
\(L(1)\) \(\approx\) \(0.8490180800 - 0.1784709181i\)
\(L(1)\) \(\approx\) \(0.8490180800 - 0.1784709181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (0.826 + 0.563i)T \)
83 \( 1 + (0.955 - 0.294i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.955 - 0.294i)T \)
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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

−25.97223214335637921651158131803, −24.76039730685925414109395918628, −23.769638017232896102670000364617, −23.36984369067496304375941039104, −22.464953133872458837869267976988, −21.44280965118943218523960937523, −20.34823336071075373655262423846, −19.149042515427463114820476647139, −18.54982697046990069554748001288, −16.99184611569668031796279155685, −16.389971221086538588660024666425, −15.76963542237296634809674284526, −14.63516731994990848363320094945, −13.667711268068855659405937418118, −12.857624181017955161297858537233, −11.76625272102015666495563200437, −10.64135311485379860570955478672, −9.086374542394553027488430555036, −8.347153814289380570655924443067, −7.19823943056259491033651553462, −6.45917672172729445897667997352, −5.15362987660245594772289771968, −3.872993551659599718829178578319, −3.32216715748406745384409780557, −0.53368027758536180237246547480, 1.51074772767116845516571734482, 3.103283480113539990436992588476, 3.72131241633415215989113819547, 5.07702457312248974704261307926, 6.17796529597864894645548550930, 7.65328433830935016919121484045, 8.82331247536525765907290762864, 9.879582081523251789465355034900, 10.79362459139067595185265584620, 12.02642678144740769236120916407, 12.39500676628227901162219069838, 13.47158110260839784822101279758, 14.8630634492655146025845351506, 15.34709759645687267159308709402, 16.551228579287005649498308029622, 18.09580885765667928409258287355, 18.75512571287771241584483629121, 19.58222481354180802459094972499, 20.38556336094520597928432567375, 21.293176891214386433719333184529, 22.4595549447622690863269061595, 23.0322772593567328660821765658, 23.629410120038388338764231716528, 25.10899427909467978167844956705, 25.90921873380537147047179834770

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