L(s) = 1 | + (−0.809 − 0.587i)3-s + i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)3-s + i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯ |
Λ(s)=(=(400s/2ΓR(s)L(s)(−0.780−0.625i)Λ(1−s)
Λ(s)=(=(400s/2ΓR(s)L(s)(−0.780−0.625i)Λ(1−s)
Degree: |
1 |
Conductor: |
400
= 24⋅52
|
Sign: |
−0.780−0.625i
|
Analytic conductor: |
1.85759 |
Root analytic conductor: |
1.85759 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ400(27,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 400, (0: ), −0.780−0.625i)
|
Particular Values
L(21) |
≈ |
0.1361366901−0.3876605351i |
L(21) |
≈ |
0.1361366901−0.3876605351i |
L(1) |
≈ |
0.6182584873−0.1588678174i |
L(1) |
≈ |
0.6182584873−0.1588678174i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
good | 3 | 1+(−0.809−0.587i)T |
| 7 | 1+iT |
| 11 | 1+(−0.951−0.309i)T |
| 13 | 1+(−0.309−0.951i)T |
| 17 | 1+(−0.587−0.809i)T |
| 19 | 1+(0.587+0.809i)T |
| 23 | 1+(−0.951−0.309i)T |
| 29 | 1+(0.587−0.809i)T |
| 31 | 1+(0.809−0.587i)T |
| 37 | 1+(−0.309−0.951i)T |
| 41 | 1+(−0.309−0.951i)T |
| 43 | 1−T |
| 47 | 1+(−0.587+0.809i)T |
| 53 | 1+(−0.809−0.587i)T |
| 59 | 1+(−0.951+0.309i)T |
| 61 | 1+(−0.951−0.309i)T |
| 67 | 1+(0.809−0.587i)T |
| 71 | 1+(−0.809−0.587i)T |
| 73 | 1+(0.951+0.309i)T |
| 79 | 1+(−0.809−0.587i)T |
| 83 | 1+(−0.809+0.587i)T |
| 89 | 1+(0.309−0.951i)T |
| 97 | 1+(0.587−0.809i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.3720672533461280870528358469, −23.63542472818526135622546664668, −23.191831653544505384439653228531, −21.95362593893559641309110104648, −21.503569060429191687406317080926, −20.394163233916193663641125164545, −19.71576913514370114108269537847, −18.361142971987097318297789200312, −17.6003097099340392480132496349, −16.84315920656300787707281475505, −15.99529340806297218743183595387, −15.255237787850840062323023212681, −14.05965441997778211690833184554, −13.16458028899732984219976182003, −12.06621163534045202674342708488, −11.17978031956272659855497629827, −10.317745644295511603909274370014, −9.696352261582907299493026715903, −8.36050226928974737188661271015, −7.09638501580768999928069440205, −6.381547009304692897532133789520, −4.97063282181268719199447103415, −4.42379567512762485348245704886, −3.19423345920826780639846317626, −1.4849072559701040194802770840,
0.26498896849805662510875437267, 2.00087878739302388283604247619, 2.95435456911883077457593818685, 4.76405917149432609078017127825, 5.57927271547639402814615044344, 6.30163783896814579888458786961, 7.65639750771766760993762249074, 8.26463209896971486001690569609, 9.72372547786250085694270842604, 10.61817581394707898206445965017, 11.68282997268812460666525049617, 12.32467516278085997151267689551, 13.19121617801598011941641921787, 14.13026200053435489231345710019, 15.59593403640639191501174605065, 15.93763944177045399014237555274, 17.19366214537328049240175689356, 18.13477357573103748280546518222, 18.479752500674838801811055360067, 19.50849662342221793682060869695, 20.64638313306267980970407361878, 21.598128580299261312136888143840, 22.53783957252884157301762065807, 22.95807458352101776818197246393, 24.28246618916523510760160366728