L(s) = 1 | + 6·2-s + 86·3-s − 804·4-s + 516·6-s + 4.80e3·7-s − 6.79e3·8-s − 1.04e4·9-s + 3.53e4·11-s − 6.91e4·12-s + 2.65e4·13-s + 2.88e4·14-s + 3.90e5·16-s + 4.63e5·17-s − 6.27e4·18-s − 9.25e5·19-s + 4.12e5·21-s + 2.11e5·22-s − 7.78e5·23-s − 5.84e5·24-s + 1.59e5·26-s − 7.43e5·27-s − 3.86e6·28-s − 1.00e7·29-s + 2.46e6·31-s + 4.00e6·32-s + 3.03e6·33-s + 2.78e6·34-s + ⋯ |
L(s) = 1 | + 0.265·2-s + 0.612·3-s − 1.57·4-s + 0.162·6-s + 0.755·7-s − 0.586·8-s − 0.531·9-s + 0.727·11-s − 0.962·12-s + 0.257·13-s + 0.200·14-s + 1.49·16-s + 1.34·17-s − 0.140·18-s − 1.62·19-s + 0.463·21-s + 0.192·22-s − 0.579·23-s − 0.359·24-s + 0.0683·26-s − 0.269·27-s − 1.18·28-s − 2.62·29-s + 0.479·31-s + 0.675·32-s + 0.445·33-s + 0.357·34-s + ⋯ |
Λ(s)=(=(30625s/2ΓC(s)2L(s)Λ(10−s)
Λ(s)=(=(30625s/2ΓC(s+9/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
30625
= 54⋅72
|
Sign: |
1
|
Analytic conductor: |
8123.64 |
Root analytic conductor: |
9.49374 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 30625, ( :9/2,9/2), 1)
|
Particular Values
L(5) |
≈ |
1.870975765 |
L(21) |
≈ |
1.870975765 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 5 | | 1 |
| 7 | C1 | (1−p4T)2 |
good | 2 | D4 | 1−3pT+105p3T2−3p10T3+p18T4 |
| 3 | D4 | 1−86T+5954pT2−86p9T3+p18T4 |
| 11 | D4 | 1−35316T+2892681078T2−35316p9T3+p18T4 |
| 13 | D4 | 1−26530T−1541163822T2−26530p9T3+p18T4 |
| 17 | D4 | 1−463920T+273833245726T2−463920p9T3+p18T4 |
| 19 | D4 | 1+925426T+858791487510T2+925426p9T3+p18T4 |
| 23 | D4 | 1+778128T+3473691840430T2+778128p9T3+p18T4 |
| 29 | D4 | 1+10003584T+52302031706070T2+10003584p9T3+p18T4 |
| 31 | D4 | 1−2467260T+49371575832542T2−2467260p9T3+p18T4 |
| 37 | D4 | 1+30735552T+484209298874630T2+30735552p9T3+p18T4 |
| 41 | D4 | 1+19103448T+602984827739166T2+19103448p9T3+p18T4 |
| 43 | D4 | 1+4065100T+797231337676374T2+4065100p9T3+p18T4 |
| 47 | D4 | 1−82195020T+3721245520696702T2−82195020p9T3+p18T4 |
| 53 | D4 | 1−55189812T+3123778356606670T2−55189812p9T3+p18T4 |
| 59 | D4 | 1+7069218T+16866494212382134T2+7069218p9T3+p18T4 |
| 61 | D4 | 1−44316386T+21654336818123658T2−44316386p9T3+p18T4 |
| 67 | D4 | 1−241921336T+59516583718815510T2−241921336p9T3+p18T4 |
| 71 | D4 | 1−206493816T+58491352612128526T2−206493816p9T3+p18T4 |
| 73 | D4 | 1−499153188T+178474458263805254T2−499153188p9T3+p18T4 |
| 79 | D4 | 1−5930824pT+239633073722978334T2−5930824p10T3+p18T4 |
| 83 | D4 | 1+444023958T+333438010641681622T2+444023958p9T3+p18T4 |
| 89 | D4 | 1−636267396T+801539802340191990T2−636267396p9T3+p18T4 |
| 97 | D4 | 1−1632716064T+2180562419544849758T2−1632716064p9T3+p18T4 |
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
−11.24736840267355414168373815103, −10.73513528579450487671986428164, −10.19767358347092717573159112406, −9.681064071062166126534867535311, −9.053738390617711546116856195206, −8.736858325823682805074417413058, −8.467463836465438712973521192822, −7.920675264502913579887568848730, −7.34633947525959988117123588851, −6.57245112476261471319148351636, −5.81019747511315724849956059286, −5.32453252970888275250377022748, −5.00044365436248585046422647362, −4.08115409205300848253758962368, −3.60850188643369384816150837197, −3.59541895133901368702273437857, −2.24912981254981469504134128471, −1.86943577280585272761878224638, −0.954087512157474385978369047606, −0.34002778790331420961708499884,
0.34002778790331420961708499884, 0.954087512157474385978369047606, 1.86943577280585272761878224638, 2.24912981254981469504134128471, 3.59541895133901368702273437857, 3.60850188643369384816150837197, 4.08115409205300848253758962368, 5.00044365436248585046422647362, 5.32453252970888275250377022748, 5.81019747511315724849956059286, 6.57245112476261471319148351636, 7.34633947525959988117123588851, 7.920675264502913579887568848730, 8.467463836465438712973521192822, 8.736858325823682805074417413058, 9.053738390617711546116856195206, 9.681064071062166126534867535311, 10.19767358347092717573159112406, 10.73513528579450487671986428164, 11.24736840267355414168373815103