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Properties

Label 1-269-269.73-r0-0-0
Degree $1$
Conductor $269$
Sign $0.986 - 0.165i$
Analytic cond. $1.24923$
Root an. cond. $1.24923$
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  + (−0.591 − 0.806i)2-s + (0.116 − 0.993i)3-s + (−0.300 + 0.953i)4-s + (0.960 + 0.277i)5-s + (−0.869 + 0.493i)6-s + (−0.513 + 0.858i)7-s + (0.946 − 0.322i)8-s + (−0.972 − 0.232i)9-s + (−0.344 − 0.938i)10-s + (−0.472 + 0.881i)11-s + (0.912 + 0.409i)12-s + (0.344 + 0.938i)13-s + (0.995 − 0.0936i)14-s + (0.388 − 0.921i)15-s + (−0.819 − 0.572i)16-s + (0.472 + 0.881i)17-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (0.116 − 0.993i)3-s + (−0.300 + 0.953i)4-s + (0.960 + 0.277i)5-s + (−0.869 + 0.493i)6-s + (−0.513 + 0.858i)7-s + (0.946 − 0.322i)8-s + (−0.972 − 0.232i)9-s + (−0.344 − 0.938i)10-s + (−0.472 + 0.881i)11-s + (0.912 + 0.409i)12-s + (0.344 + 0.938i)13-s + (0.995 − 0.0936i)14-s + (0.388 − 0.921i)15-s + (−0.819 − 0.572i)16-s + (0.472 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9071759229 - 0.07544535978i\)
\(L(\frac12)\) \(\approx\) \(0.9071759229 - 0.07544535978i\)
\(L(1)\) \(\approx\) \(0.8197639150 - 0.2342779810i\)
\(L(1)\) \(\approx\) \(0.8197639150 - 0.2342779810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.591 - 0.806i)T \)
3 \( 1 + (0.116 - 0.993i)T \)
5 \( 1 + (0.960 + 0.277i)T \)
7 \( 1 + (-0.513 + 0.858i)T \)
11 \( 1 + (-0.472 + 0.881i)T \)
13 \( 1 + (0.344 + 0.938i)T \)
17 \( 1 + (0.472 + 0.881i)T \)
19 \( 1 + (0.209 - 0.977i)T \)
23 \( 1 + (-0.116 + 0.993i)T \)
29 \( 1 + (-0.845 + 0.533i)T \)
31 \( 1 + (0.553 + 0.833i)T \)
37 \( 1 + (-0.628 + 0.777i)T \)
41 \( 1 + (0.591 - 0.806i)T \)
43 \( 1 + (0.845 - 0.533i)T \)
47 \( 1 + (0.163 - 0.986i)T \)
53 \( 1 + (-0.0234 + 0.999i)T \)
59 \( 1 + (0.300 - 0.953i)T \)
61 \( 1 + (0.255 - 0.966i)T \)
67 \( 1 + (-0.300 - 0.953i)T \)
71 \( 1 + (-0.995 - 0.0936i)T \)
73 \( 1 + (0.982 - 0.186i)T \)
79 \( 1 + (-0.990 + 0.140i)T \)
83 \( 1 + (-0.892 - 0.451i)T \)
89 \( 1 + (0.731 - 0.681i)T \)
97 \( 1 + (0.255 + 0.966i)T \)
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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

−25.9425522059111824907689252196, −25.09792958175025062122822996739, −24.25610976107433455212833955813, −22.803221477418266503526398369367, −22.55493236613027691022397670261, −20.900746919974623742593504408251, −20.5547495302188207132819645102, −19.297620128261048722302917929437, −18.2037309265672419913230120088, −17.2280753164401935871648453496, −16.37183024685441877176486974742, −16.043455512164590865475039406727, −14.63879573540897738763112486746, −13.919398019479437588841818699360, −13.077526115933264004512106157729, −11.04015640238328174537844834825, −10.19816450912317370023330107763, −9.691176913157845471709673065249, −8.600184512024585288794767936592, −7.640199968641919378527473714855, −6.03752697588421019816937144873, −5.55821878314840511944663027859, −4.25665098222938459919068203702, −2.80767089442321253034317781682, −0.774994579223579204389035367273, 1.59607595796998008874389217380, 2.26977498801456817474645536266, 3.3451670991737547721616148291, 5.27764004435506030866746535097, 6.52294599097868048026106297810, 7.42074213933108623146870437632, 8.78878074293241650365412619283, 9.3880614061829454120848827075, 10.520548520959553131120583111589, 11.70201934429580379543229569183, 12.567268654535154555008464231264, 13.26352744407788203684185380994, 14.17595597075655754486768867821, 15.589125123202081697714390618248, 17.06122606160546494850878499906, 17.68265325396364621803859962713, 18.555737009524696769230403081500, 19.07477565475323127085333269201, 20.08760733394982022705243357229, 21.17296182398973999232608344411, 21.88868587483187372089227197076, 22.84962658414612584711582765432, 23.9963940976442645272428382327, 25.23818346005540214576767220539, 25.8428603133068236739584938011

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