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Properties

Label 1-269-269.51-r0-0-0
Degree 11
Conductor 269269
Sign 0.847−0.530i0.847 - 0.530i
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  + (0.300 − 0.953i)2-s + (0.972 + 0.232i)3-s + (−0.819 − 0.572i)4-s + (0.845 + 0.533i)5-s + (0.513 − 0.858i)6-s + (0.472 + 0.881i)7-s + (−0.792 + 0.610i)8-s + (0.892 + 0.451i)9-s + (0.762 − 0.646i)10-s + (−0.553 − 0.833i)11-s + (−0.664 − 0.747i)12-s + (−0.762 + 0.646i)13-s + (0.982 − 0.186i)14-s + (0.698 + 0.715i)15-s + (0.344 + 0.938i)16-s + (0.553 − 0.833i)17-s + ⋯
L(s)  = 1  + (0.300 − 0.953i)2-s + (0.972 + 0.232i)3-s + (−0.819 − 0.572i)4-s + (0.845 + 0.533i)5-s + (0.513 − 0.858i)6-s + (0.472 + 0.881i)7-s + (−0.792 + 0.610i)8-s + (0.892 + 0.451i)9-s + (0.762 − 0.646i)10-s + (−0.553 − 0.833i)11-s + (−0.664 − 0.747i)12-s + (−0.762 + 0.646i)13-s + (0.982 − 0.186i)14-s + (0.698 + 0.715i)15-s + (0.344 + 0.938i)16-s + (0.553 − 0.833i)17-s + ⋯

Functional equation

Λ(s)=(269s/2 ΓR(s) L(s)=((0.847−0.530i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(269s/2 ΓR(s) L(s)=((0.847−0.530i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 269269
Sign: 0.847−0.530i0.847 - 0.530i
Analytic conductor: 1.249231.24923
Root analytic conductor: 1.249231.24923
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ269(51,⋅)\chi_{269} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 269, (0: ), 0.847−0.530i)(1,\ 269,\ (0:\ ),\ 0.847 - 0.530i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.983027836−0.5698697280i1.983027836 - 0.5698697280i
L(12)L(\frac12) ≈\approx 1.983027836−0.5698697280i1.983027836 - 0.5698697280i
L(1)L(1) ≈\approx 1.632232028−0.4458896832i1.632232028 - 0.4458896832i
L(1)L(1) ≈\approx 1.632232028−0.4458896832i1.632232028 - 0.4458896832i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad269 1 1
good2 1+(0.300−0.953i)T 1 + (0.300 - 0.953i)T
3 1+(0.972+0.232i)T 1 + (0.972 + 0.232i)T
5 1+(0.845+0.533i)T 1 + (0.845 + 0.533i)T
7 1+(0.472+0.881i)T 1 + (0.472 + 0.881i)T
11 1+(−0.553−0.833i)T 1 + (-0.553 - 0.833i)T
13 1+(−0.762+0.646i)T 1 + (-0.762 + 0.646i)T
17 1+(0.553−0.833i)T 1 + (0.553 - 0.833i)T
19 1+(0.912+0.409i)T 1 + (0.912 + 0.409i)T
23 1+(−0.972−0.232i)T 1 + (-0.972 - 0.232i)T
29 1+(−0.430+0.902i)T 1 + (-0.430 + 0.902i)T
31 1+(0.388−0.921i)T 1 + (0.388 - 0.921i)T
37 1+(−0.209−0.977i)T 1 + (-0.209 - 0.977i)T
41 1+(−0.300−0.953i)T 1 + (-0.300 - 0.953i)T
43 1+(0.430−0.902i)T 1 + (0.430 - 0.902i)T
47 1+(−0.946−0.322i)T 1 + (-0.946 - 0.322i)T
53 1+(−0.998−0.0468i)T 1 + (-0.998 - 0.0468i)T
59 1+(0.819+0.572i)T 1 + (0.819 + 0.572i)T
61 1+(−0.869−0.493i)T 1 + (-0.869 - 0.493i)T
67 1+(−0.819+0.572i)T 1 + (-0.819 + 0.572i)T
71 1+(−0.982−0.186i)T 1 + (-0.982 - 0.186i)T
73 1+(0.930−0.366i)T 1 + (0.930 - 0.366i)T
79 1+(0.960−0.277i)T 1 + (0.960 - 0.277i)T
83 1+(−0.591−0.806i)T 1 + (-0.591 - 0.806i)T
89 1+(0.0702−0.997i)T 1 + (0.0702 - 0.997i)T
97 1+(−0.869+0.493i)T 1 + (-0.869 + 0.493i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−25.80506512388070725215033407649, −24.889524576002351870532280166855, −24.238647575115563035497508381038, −23.492306692015279875669210745616, −22.252034201354411879218968003841, −21.1941934648649261552865278540, −20.53178594621734939817020413, −19.57446174339617115648484427745, −18.00042192652638632721239380402, −17.65381841500553759971863430341, −16.58770266059937390059399851710, −15.43460713916999561548651077078, −14.57812252291886622969298381854, −13.799181181775991177049171786198, −13.08550488793662437629492233870, −12.27670272960367465953817901686, −10.053172551586873700360444461732, −9.62596526783379603657351829020, −8.12416330040067951772495561469, −7.73796131493131844294189851305, −6.56102172062576049398598089272, −5.188609211825938204812412730913, −4.35196950281467883088229989606, −2.964168071046903811004513110210, −1.4401157635546919059675449928, 1.801903531444037707061142639481, 2.55644289442653421219470236248, 3.44796796367567417041295400887, 4.95967189524655384407635084229, 5.794463372722888753285849218909, 7.55432078902505653822739505197, 8.81444671282634704393162349479, 9.55451610684322896383520909204, 10.36186516735103443803689122180, 11.51248969040187540853238219022, 12.54180871316209280095134928507, 13.82164738389261435286294783901, 14.142413442963953056949872227939, 15.02136109469576393302150385020, 16.2772976089229946006266230493, 17.92568071777364373041576170537, 18.610830269393209880789885470550, 19.17525897695878182859461762996, 20.52683836838997766314912458683, 21.06650693198344177801207987948, 21.885281475947592457405808332155, 22.38218545021702960861861100309, 24.08163023924403826726271951897, 24.65835139050537514707054909738, 25.84410932514193401136920616198

Graph of the ZZ-function along the critical line