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Properties

Label 1-269-269.67-r0-0-0
Degree 11
Conductor 269269
Sign 0.0121−0.999i0.0121 - 0.999i
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  + (−0.998 + 0.0468i)2-s + (−0.388 − 0.921i)3-s + (0.995 − 0.0936i)4-s + (−0.946 + 0.322i)5-s + (0.430 + 0.902i)6-s + (−0.628 + 0.777i)7-s + (−0.990 + 0.140i)8-s + (−0.698 + 0.715i)9-s + (0.930 − 0.366i)10-s + (−0.209 + 0.977i)11-s + (−0.472 − 0.881i)12-s + (0.930 − 0.366i)13-s + (0.591 − 0.806i)14-s + (0.664 + 0.747i)15-s + (0.982 − 0.186i)16-s + (−0.209 − 0.977i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0468i)2-s + (−0.388 − 0.921i)3-s + (0.995 − 0.0936i)4-s + (−0.946 + 0.322i)5-s + (0.430 + 0.902i)6-s + (−0.628 + 0.777i)7-s + (−0.990 + 0.140i)8-s + (−0.698 + 0.715i)9-s + (0.930 − 0.366i)10-s + (−0.209 + 0.977i)11-s + (−0.472 − 0.881i)12-s + (0.930 − 0.366i)13-s + (0.591 − 0.806i)14-s + (0.664 + 0.747i)15-s + (0.982 − 0.186i)16-s + (−0.209 − 0.977i)17-s + ⋯

Functional equation

Λ(s)=(269s/2 ΓR(s) L(s)=((0.0121−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(269s/2 ΓR(s) L(s)=((0.0121−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 269269
Sign: 0.0121−0.999i0.0121 - 0.999i
Analytic conductor: 1.249231.24923
Root analytic conductor: 1.249231.24923
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ269(67,⋅)\chi_{269} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 269, (0: ), 0.0121−0.999i)(1,\ 269,\ (0:\ ),\ 0.0121 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.2660318582−0.2692823239i0.2660318582 - 0.2692823239i
L(12)L(\frac12) ≈\approx 0.2660318582−0.2692823239i0.2660318582 - 0.2692823239i
L(1)L(1) ≈\approx 0.4516769478−0.1111272597i0.4516769478 - 0.1111272597i
L(1)L(1) ≈\approx 0.4516769478−0.1111272597i0.4516769478 - 0.1111272597i

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+(−0.998+0.0468i)T 1 + (-0.998 + 0.0468i)T
3 1+(−0.388−0.921i)T 1 + (-0.388 - 0.921i)T
5 1+(−0.946+0.322i)T 1 + (-0.946 + 0.322i)T
7 1+(−0.628+0.777i)T 1 + (-0.628 + 0.777i)T
11 1+(−0.209+0.977i)T 1 + (-0.209 + 0.977i)T
13 1+(0.930−0.366i)T 1 + (0.930 - 0.366i)T
17 1+(−0.209−0.977i)T 1 + (-0.209 - 0.977i)T
19 1+(0.513−0.858i)T 1 + (0.513 - 0.858i)T
23 1+(−0.388−0.921i)T 1 + (-0.388 - 0.921i)T
29 1+(0.792+0.610i)T 1 + (0.792 + 0.610i)T
31 1+(−0.912−0.409i)T 1 + (-0.912 - 0.409i)T
37 1+(−0.869−0.493i)T 1 + (-0.869 - 0.493i)T
41 1+(−0.998−0.0468i)T 1 + (-0.998 - 0.0468i)T
43 1+(0.792+0.610i)T 1 + (0.792 + 0.610i)T
47 1+(0.0702−0.997i)T 1 + (0.0702 - 0.997i)T
53 1+(−0.972−0.232i)T 1 + (-0.972 - 0.232i)T
59 1+(0.995−0.0936i)T 1 + (0.995 - 0.0936i)T
61 1+(0.845−0.533i)T 1 + (0.845 - 0.533i)T
67 1+(0.995+0.0936i)T 1 + (0.995 + 0.0936i)T
71 1+(0.591+0.806i)T 1 + (0.591 + 0.806i)T
73 1+(−0.300−0.953i)T 1 + (-0.300 - 0.953i)T
79 1+(0.163−0.986i)T 1 + (0.163 - 0.986i)T
83 1+(−0.0234−0.999i)T 1 + (-0.0234 - 0.999i)T
89 1+(0.344−0.938i)T 1 + (0.344 - 0.938i)T
97 1+(0.845+0.533i)T 1 + (0.845 + 0.533i)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

−26.29049197287988353458145285680, −25.515555018524041782176641486509, −23.91560955058214518985487466306, −23.541753727645670522877021209823, −22.291569664841842741649331560864, −21.14947657203253688118988754340, −20.44332074188987432595877921062, −19.54842295732437059352832228345, −18.81703169450446929192794876042, −17.47525824961093541120135916530, −16.59657980946932135579035688621, −16.0233524737939632147748216733, −15.46135130071314424816123824224, −13.99670226665764097387166087675, −12.482241582485250279620488168652, −11.41789465960254382318627603128, −10.78852817715283444922671138741, −9.88883082358845412340059990821, −8.76487002625044620826026298028, −8.02271557803195576191482844275, −6.66642309413635464153232489205, −5.652504630818704886935108109485, −3.85445822722085203432665035939, −3.40871140933868331404085873444, −1.06781647475174499587663751936, 0.456980105598020849825756014113, 2.16344020879188144888714692975, 3.15984719537469752959988766398, 5.2218093492833866026359687898, 6.56471819770903681639818051577, 7.12422874929127933564400506639, 8.16634066680219924188195733575, 9.056838830592042168307461079850, 10.421630928503233912027174403823, 11.411331427285473280477803391660, 12.11941804334929117332104257323, 12.96947512518578760014997028784, 14.56864696567333721723882000842, 15.7949712492487485285292596752, 16.10358057454411214960494961669, 17.60611830823039233590649120083, 18.32720895861241724966985383233, 18.80445881222443240741026753102, 19.871961066948446625543838276416, 20.414061333730821330315706938178, 22.15884762765744999104766142123, 22.95918295397949485084879010646, 23.83380602493245292235248863184, 24.80501687121258701706235194340, 25.5672656531228667530342647431

Graph of the ZZ-function along the critical line