L(s) = 1 | − 3·2-s + 4-s − 10·5-s − 3·8-s + 30·10-s − 62·11-s + 6·13-s + 9·16-s + 40·17-s + 122·19-s − 10·20-s + 186·22-s − 16·23-s + 75·25-s − 18·26-s − 352·29-s − 66·31-s + 165·32-s − 120·34-s − 188·37-s − 366·38-s + 30·40-s + 16·41-s − 396·43-s − 62·44-s + 48·46-s − 188·47-s + ⋯ |
L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.894·5-s − 0.132·8-s + 0.948·10-s − 1.69·11-s + 0.128·13-s + 9/64·16-s + 0.570·17-s + 1.47·19-s − 0.111·20-s + 1.80·22-s − 0.145·23-s + 3/5·25-s − 0.135·26-s − 2.25·29-s − 0.382·31-s + 0.911·32-s − 0.605·34-s − 0.835·37-s − 1.56·38-s + 0.118·40-s + 0.0609·41-s − 1.40·43-s − 0.212·44-s + 0.153·46-s − 0.583·47-s + ⋯ |
Λ(s)=(=(4862025s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(4862025s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
4862025
= 34⋅52⋅74
|
Sign: |
1
|
Analytic conductor: |
16925.8 |
Root analytic conductor: |
11.4061 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 4862025, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
0.5902238494 |
L(21) |
≈ |
0.5902238494 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C1 | (1+pT)2 |
| 7 | | 1 |
good | 2 | D4 | 1+3T+p3T2+3p3T3+p6T4 |
| 11 | D4 | 1+62T+3582T2+62p3T3+p6T4 |
| 13 | D4 | 1−6T+3378T2−6p3T3+p6T4 |
| 17 | D4 | 1−40T+8750T2−40p3T3+p6T4 |
| 19 | D4 | 1−122T+17398T2−122p3T3+p6T4 |
| 23 | D4 | 1+16T+34pT2+16p3T3+p6T4 |
| 29 | D4 | 1+352T+78278T2+352p3T3+p6T4 |
| 31 | D4 | 1+66T+45870T2+66p3T3+p6T4 |
| 37 | D4 | 1+188T+44542T2+188p3T3+p6T4 |
| 41 | D4 | 1−16T+18350T2−16p3T3+p6T4 |
| 43 | D4 | 1+396T+2226pT2+396p3T3+p6T4 |
| 47 | D4 | 1+4pT+15582T2+4p4T3+p6T4 |
| 53 | D4 | 1+982T+504354T2+982p3T3+p6T4 |
| 59 | D4 | 1−516T+418118T2−516p3T3+p6T4 |
| 61 | D4 | 1−880T+575238T2−880p3T3+p6T4 |
| 67 | D4 | 1+356T+100374T2+356p3T3+p6T4 |
| 71 | D4 | 1+310T+664038T2+310p3T3+p6T4 |
| 73 | D4 | 1+326T+789802T2+326p3T3+p6T4 |
| 79 | D4 | 1−1832T+1824478T2−1832p3T3+p6T4 |
| 83 | D4 | 1+680T+744870T2+680p3T3+p6T4 |
| 89 | D4 | 1−796T+411158T2−796p3T3+p6T4 |
| 97 | D4 | 1−670T+1145410T2−670p3T3+p6T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.765023127653518868449475180324, −8.552390438446042517550388443818, −8.011307591654896696318463065031, −7.928121915751284233212196480082, −7.43039046950796777063564665707, −7.28162703920366389412844994007, −6.77500121526243522454268513281, −6.10789216281293128148667186853, −5.76951821820016139012997102054, −5.29085316505930329359099282207, −4.85458081491807854820364249910, −4.75093120025812212571984131523, −3.71303461658186779984528906303, −3.50589041345024107644173498813, −3.21303841623059107067077121610, −2.55841064261338600173888301568, −1.94340258328469866153911550303, −1.41669654627431476667269737316, −0.52963839797465127935562734935, −0.35838979906258974904439614630,
0.35838979906258974904439614630, 0.52963839797465127935562734935, 1.41669654627431476667269737316, 1.94340258328469866153911550303, 2.55841064261338600173888301568, 3.21303841623059107067077121610, 3.50589041345024107644173498813, 3.71303461658186779984528906303, 4.75093120025812212571984131523, 4.85458081491807854820364249910, 5.29085316505930329359099282207, 5.76951821820016139012997102054, 6.10789216281293128148667186853, 6.77500121526243522454268513281, 7.28162703920366389412844994007, 7.43039046950796777063564665707, 7.928121915751284233212196480082, 8.011307591654896696318463065031, 8.552390438446042517550388443818, 8.765023127653518868449475180324