L(s) = 1 | − 8·2-s + 48·4-s + 54·5-s − 74·7-s − 256·8-s − 432·10-s + 78·11-s − 1.10e3·13-s + 592·14-s + 1.28e3·16-s + 492·17-s − 1.64e3·19-s + 2.59e3·20-s − 624·22-s + 5.53e3·23-s − 3.19e3·25-s + 8.84e3·26-s − 3.55e3·28-s + 3.89e3·29-s − 4.71e3·31-s − 6.14e3·32-s − 3.93e3·34-s − 3.99e3·35-s − 4.79e3·37-s + 1.31e4·38-s − 1.38e4·40-s − 1.53e4·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.965·5-s − 0.570·7-s − 1.41·8-s − 1.36·10-s + 0.194·11-s − 1.81·13-s + 0.807·14-s + 5/4·16-s + 0.412·17-s − 1.04·19-s + 1.44·20-s − 0.274·22-s + 2.18·23-s − 1.02·25-s + 2.56·26-s − 0.856·28-s + 0.859·29-s − 0.881·31-s − 1.06·32-s − 0.583·34-s − 0.551·35-s − 0.575·37-s + 1.47·38-s − 1.36·40-s − 1.42·41-s + ⋯ |
Λ(s)=(=(26244s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(26244s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
26244
= 22⋅38
|
Sign: |
1
|
Analytic conductor: |
675.073 |
Root analytic conductor: |
5.09727 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 26244, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1+p2T)2 |
| 3 | | 1 |
good | 5 | D4 | 1−54T+1223pT2−54p5T3+p10T4 |
| 7 | D4 | 1+74T+34767T2+74p5T3+p10T4 |
| 11 | D4 | 1−78T−75761T2−78p5T3+p10T4 |
| 13 | D4 | 1+1106T+961995T2+1106p5T3+p10T4 |
| 17 | D4 | 1−492T+991654T2−492p5T3+p10T4 |
| 19 | D4 | 1+1640T+2303382T2+1640p5T3+p10T4 |
| 23 | D4 | 1−5538T+20405047T2−5538p5T3+p10T4 |
| 29 | D4 | 1−3894T+41491891T2−3894p5T3+p10T4 |
| 31 | D4 | 1+4718T+21950583T2+4718p5T3+p10T4 |
| 37 | D4 | 1+4796T−2933298T2+4796p5T3+p10T4 |
| 41 | D4 | 1+15354T+290606395T2+15354p5T3+p10T4 |
| 43 | D4 | 1+32858T+548530503T2+32858p5T3+p10T4 |
| 47 | D4 | 1+24954T+600973327T2+24954p5T3+p10T4 |
| 53 | D4 | 1+16332T+896230798T2+16332p5T3+p10T4 |
| 59 | D4 | 1+21966T+1529583151T2+21966p5T3+p10T4 |
| 61 | D4 | 1+50pT+1583245203T2+50p6T3+p10T4 |
| 67 | D4 | 1+36758T+36341997pT2+36758p5T3+p10T4 |
| 71 | D4 | 1+73848T+4273554814T2+73848p5T3+p10T4 |
| 73 | D4 | 1+51188T+63530214pT2+51188p5T3+p10T4 |
| 79 | D4 | 1−14926T+6101841783T2−14926p5T3+p10T4 |
| 83 | D4 | 1+90762T+9932848903T2+90762p5T3+p10T4 |
| 89 | D4 | 1+9300T+3189231862T2+9300p5T3+p10T4 |
| 97 | D4 | 1−30262T+8068652859T2−30262p5T3+p10T4 |
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
−11.61298425393125072477097267809, −11.11972086089884166513777481985, −10.26092463063207030354359232201, −10.18230866603900699321846591041, −9.591474388889140797939902171336, −9.442414125922546825429919912384, −8.540636407442362719359062707768, −8.402470482006938691222235214975, −7.37433833507661145708285367591, −7.13806529114169311531517972716, −6.46618867659138973536813763792, −6.10113001626321159582088030157, −5.09426717300798439183170142182, −4.78352265788203075760844385039, −3.21747633950733596213687698259, −2.97438701421957663850008540384, −1.77957628053872271084307267445, −1.62572664444521784498626883956, 0, 0,
1.62572664444521784498626883956, 1.77957628053872271084307267445, 2.97438701421957663850008540384, 3.21747633950733596213687698259, 4.78352265788203075760844385039, 5.09426717300798439183170142182, 6.10113001626321159582088030157, 6.46618867659138973536813763792, 7.13806529114169311531517972716, 7.37433833507661145708285367591, 8.402470482006938691222235214975, 8.540636407442362719359062707768, 9.442414125922546825429919912384, 9.591474388889140797939902171336, 10.18230866603900699321846591041, 10.26092463063207030354359232201, 11.11972086089884166513777481985, 11.61298425393125072477097267809