L(s) = 1 | − 6·2-s + 14·4-s + 6·5-s − 36·10-s − 48·11-s − 52·13-s − 84·16-s + 30·17-s − 64·19-s + 84·20-s + 288·22-s − 60·23-s − 175·25-s + 312·26-s − 360·29-s + 140·31-s + 216·32-s − 180·34-s − 230·37-s + 384·38-s + 234·41-s − 938·43-s − 672·44-s + 360·46-s + 618·47-s + 1.05e3·50-s − 728·52-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 7/4·4-s + 0.536·5-s − 1.13·10-s − 1.31·11-s − 1.10·13-s − 1.31·16-s + 0.428·17-s − 0.772·19-s + 0.939·20-s + 2.79·22-s − 0.543·23-s − 7/5·25-s + 2.35·26-s − 2.30·29-s + 0.811·31-s + 1.19·32-s − 0.907·34-s − 1.02·37-s + 1.63·38-s + 0.891·41-s − 3.32·43-s − 2.30·44-s + 1.15·46-s + 1.91·47-s + 2.96·50-s − 1.94·52-s + ⋯ |
Λ(s)=(=(1750329s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(1750329s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1750329
= 36⋅74
|
Sign: |
1
|
Analytic conductor: |
6093.28 |
Root analytic conductor: |
8.83513 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 1750329, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
0.3280486492 |
L(21) |
≈ |
0.3280486492 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 7 | | 1 |
good | 2 | D4 | 1+3pT+11pT2+3p4T3+p6T4 |
| 5 | D4 | 1−6T+211T2−6p3T3+p6T4 |
| 11 | D4 | 1+48T+2470T2+48p3T3+p6T4 |
| 13 | D4 | 1+4pT+3342T2+4p4T3+p6T4 |
| 17 | D4 | 1−30T+7699T2−30p3T3+p6T4 |
| 19 | D4 | 1+64T+3942T2+64p3T3+p6T4 |
| 23 | D4 | 1+60T+494pT2+60p3T3+p6T4 |
| 29 | D4 | 1+360T+77290T2+360p3T3+p6T4 |
| 31 | D4 | 1−140T+48930T2−140p3T3+p6T4 |
| 37 | D4 | 1+230T+98979T2+230p3T3+p6T4 |
| 41 | D4 | 1−234T+26683T2−234p3T3+p6T4 |
| 43 | D4 | 1+938T+378543T2+938p3T3+p6T4 |
| 47 | D4 | 1−618T+210199T2−618p3T3+p6T4 |
| 53 | D4 | 1−420T+64606T2−420p3T3+p6T4 |
| 59 | D4 | 1−282T+411439T2−282p3T3+p6T4 |
| 61 | D4 | 1−32T+205386T2−32p3T3+p6T4 |
| 67 | D4 | 1−544T+535542T2−544p3T3+p6T4 |
| 71 | D4 | 1−504T+736126T2−504p3T3+p6T4 |
| 73 | D4 | 1−764T+653958T2−764p3T3+p6T4 |
| 79 | D4 | 1−238T+125439T2−238p3T3+p6T4 |
| 83 | D4 | 1+522T+651823T2+522p3T3+p6T4 |
| 89 | D4 | 1−708T+1163542T2−708p3T3+p6T4 |
| 97 | D4 | 1+664T+1779618T2+664p3T3+p6T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.454797620452498735747298468023, −9.201469306942394824899384266169, −8.656354916581939464317960395147, −8.255240893443178631128607904750, −7.908415229351685857426441268681, −7.78191408827438268485925146769, −7.23983179163634404191904145114, −6.85153724283759710360390840084, −6.37468083428123473232078564898, −5.61261705391939498576129662142, −5.31722524763747337099803341728, −5.15649706773807242275127136286, −4.14894315093212786988719565516, −3.97084032343759934370877409845, −3.11144669706821286879265207432, −2.42981308880365942153631857327, −1.85913980337783281874636711339, −1.81117132762125842445054730865, −0.61833833769799951755914455817, −0.31136106903687004700543830908,
0.31136106903687004700543830908, 0.61833833769799951755914455817, 1.81117132762125842445054730865, 1.85913980337783281874636711339, 2.42981308880365942153631857327, 3.11144669706821286879265207432, 3.97084032343759934370877409845, 4.14894315093212786988719565516, 5.15649706773807242275127136286, 5.31722524763747337099803341728, 5.61261705391939498576129662142, 6.37468083428123473232078564898, 6.85153724283759710360390840084, 7.23983179163634404191904145114, 7.78191408827438268485925146769, 7.908415229351685857426441268681, 8.255240893443178631128607904750, 8.656354916581939464317960395147, 9.201469306942394824899384266169, 9.454797620452498735747298468023