Properties

Label 1-269-269.199-r0-0-0
Degree 11
Conductor 269269
Sign 0.986+0.165i0.986 + 0.165i
Analytic cond. 1.249231.24923
Root an. cond. 1.249231.24923
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(269s/2ΓR(s)L(s)=((0.986+0.165i)Λ(1s)\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}
Λ(s)=(269s/2ΓR(s)L(s)=((0.986+0.165i)Λ(1s)\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: 11
Conductor: 269269
Sign: 0.986+0.165i0.986 + 0.165i
Analytic conductor: 1.249231.24923
Root analytic conductor: 1.249231.24923
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ269(199,)\chi_{269} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 269, (0: ), 0.986+0.165i)(1,\ 269,\ (0:\ ),\ 0.986 + 0.165i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.9071759229+0.07544535978i0.9071759229 + 0.07544535978i
L(12)L(\frac12) \approx 0.9071759229+0.07544535978i0.9071759229 + 0.07544535978i
L(1)L(1) \approx 0.8197639150+0.2342779810i0.8197639150 + 0.2342779810i
L(1)L(1) \approx 0.8197639150+0.2342779810i0.8197639150 + 0.2342779810i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad269 1 1
good2 1+(0.591+0.806i)T 1 + (-0.591 + 0.806i)T
3 1+(0.116+0.993i)T 1 + (0.116 + 0.993i)T
5 1+(0.9600.277i)T 1 + (0.960 - 0.277i)T
7 1+(0.5130.858i)T 1 + (-0.513 - 0.858i)T
11 1+(0.4720.881i)T 1 + (-0.472 - 0.881i)T
13 1+(0.3440.938i)T 1 + (0.344 - 0.938i)T
17 1+(0.4720.881i)T 1 + (0.472 - 0.881i)T
19 1+(0.209+0.977i)T 1 + (0.209 + 0.977i)T
23 1+(0.1160.993i)T 1 + (-0.116 - 0.993i)T
29 1+(0.8450.533i)T 1 + (-0.845 - 0.533i)T
31 1+(0.5530.833i)T 1 + (0.553 - 0.833i)T
37 1+(0.6280.777i)T 1 + (-0.628 - 0.777i)T
41 1+(0.591+0.806i)T 1 + (0.591 + 0.806i)T
43 1+(0.845+0.533i)T 1 + (0.845 + 0.533i)T
47 1+(0.163+0.986i)T 1 + (0.163 + 0.986i)T
53 1+(0.02340.999i)T 1 + (-0.0234 - 0.999i)T
59 1+(0.300+0.953i)T 1 + (0.300 + 0.953i)T
61 1+(0.255+0.966i)T 1 + (0.255 + 0.966i)T
67 1+(0.300+0.953i)T 1 + (-0.300 + 0.953i)T
71 1+(0.995+0.0936i)T 1 + (-0.995 + 0.0936i)T
73 1+(0.982+0.186i)T 1 + (0.982 + 0.186i)T
79 1+(0.9900.140i)T 1 + (-0.990 - 0.140i)T
83 1+(0.892+0.451i)T 1 + (-0.892 + 0.451i)T
89 1+(0.731+0.681i)T 1 + (0.731 + 0.681i)T
97 1+(0.2550.966i)T 1 + (0.255 - 0.966i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

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

Graph of the ZZ-function along the critical line