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Properties

Label 1-261-261.182-r0-0-0
Degree $1$
Conductor $261$
Sign $0.0681 + 0.997i$
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  + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (0.365 + 0.930i)5-s + (0.955 − 0.294i)7-s + (0.433 − 0.900i)8-s + (−0.974 + 0.222i)10-s + (0.563 − 0.826i)11-s + (−0.0747 + 0.997i)13-s + (0.149 + 0.988i)14-s + (0.826 + 0.563i)16-s − i·17-s + (0.974 − 0.222i)19-s + (−0.0747 − 0.997i)20-s + (0.733 + 0.680i)22-s + (0.988 − 0.149i)23-s + ⋯
L(s)  = 1  + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (0.365 + 0.930i)5-s + (0.955 − 0.294i)7-s + (0.433 − 0.900i)8-s + (−0.974 + 0.222i)10-s + (0.563 − 0.826i)11-s + (−0.0747 + 0.997i)13-s + (0.149 + 0.988i)14-s + (0.826 + 0.563i)16-s − i·17-s + (0.974 − 0.222i)19-s + (−0.0747 − 0.997i)20-s + (0.733 + 0.680i)22-s + (0.988 − 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0681 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0681 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.0681 + 0.997i$
Analytic conductor: \(1.21207\)
Root analytic conductor: \(1.21207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (0:\ ),\ 0.0681 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9186263382 + 0.8580167332i\)
\(L(\frac12)\) \(\approx\) \(0.9186263382 + 0.8580167332i\)
\(L(1)\) \(\approx\) \(0.9374944365 + 0.5635234200i\)
\(L(1)\) \(\approx\) \(0.9374944365 + 0.5635234200i\)

Euler product

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

−25.60001697968046386975511738650, −24.79236696586311125093610190440, −23.79764635803601219335856994726, −22.698958770003736280303940557412, −21.788083477847542894568529478525, −20.88224175254605627765847466954, −20.31329569758509826324811073042, −19.48233805217454505349935972977, −18.12488155151518044445034740522, −17.5541305400047868642650322399, −16.7829300068893227389107548998, −15.19601040309247937312582807238, −14.26668599506901788365047020049, −13.06850175817375204596788314473, −12.41950650712988131279852868647, −11.50519124397601287573933520080, −10.394749597873551912899215557009, −9.397996256246822635759878245893, −8.553428002922181243306189389029, −7.603189525044815034140422439616, −5.55826348167781197612314018247, −4.85417471341481574244862371879, −3.66915725637122809773470875399, −2.070144971946592612821112568366, −1.20837014936185318046496542104, 1.38018074412927636775605797659, 3.21250345554380714318748928091, 4.578146757342181885305152072364, 5.62840190508624208301074733898, 6.82603963574205968775789824759, 7.40886419628671965692351268642, 8.74145416610245313643249862948, 9.58296007610092514786606963446, 10.86183344401468530024706291004, 11.68256026305705195485133410322, 13.510901317642350410125507053698, 14.13913712488265278019635514497, 14.68610510914800875794596021284, 15.90759316541239634367853494913, 16.85787532278657128386003583478, 17.70683284963109361897063672692, 18.53235923787930325015280833095, 19.240795341683829363445398080877, 20.75334147436543929492126747910, 21.84228990205923500595386451838, 22.474091304800720645519160838766, 23.57633578376965232644053633319, 24.343076899370021860015274518132, 25.1293170906004705823393663841, 26.139277159897817653457678748813

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