L(s) = 1 | + 4·2-s + 10·4-s − 28·7-s + 17·8-s − 100·11-s − 44·13-s − 112·14-s − 10·16-s − 53·17-s − 29·19-s − 400·22-s + 295·23-s − 176·26-s − 280·28-s − 129·29-s + 114·31-s − 212·32-s − 212·34-s − 403·37-s − 116·38-s − 671·41-s + 411·43-s − 1.00e3·44-s + 1.18e3·46-s − 8·47-s + 490·49-s − 440·52-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 5/4·4-s − 1.51·7-s + 0.751·8-s − 2.74·11-s − 0.938·13-s − 2.13·14-s − 0.156·16-s − 0.756·17-s − 0.350·19-s − 3.87·22-s + 2.67·23-s − 1.32·26-s − 1.88·28-s − 0.826·29-s + 0.660·31-s − 1.17·32-s − 1.06·34-s − 1.79·37-s − 0.495·38-s − 2.55·41-s + 1.45·43-s − 3.42·44-s + 3.78·46-s − 0.0248·47-s + 10/7·49-s − 1.17·52-s + ⋯ |
Λ(s)=(=((38⋅58⋅74)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((38⋅58⋅74)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
38⋅58⋅74
|
Sign: |
1
|
Analytic conductor: |
7.45738×107 |
Root analytic conductor: |
9.63991 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 38⋅58⋅74, ( :3/2,3/2,3/2,3/2), 1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | | 1 |
| 7 | C1 | (1+pT)4 |
good | 2 | C2≀S4 | 1−p2T+3pT2−T3+11pT4−p3T5+3p7T6−p11T7+p12T8 |
| 11 | C2≀S4 | 1+100T+4738T2+106848T3+2242403T4+106848p3T5+4738p6T6+100p9T7+p12T8 |
| 13 | C2≀S4 | 1+44T+4820T2+154316T3+10485014T4+154316p3T5+4820p6T6+44p9T7+p12T8 |
| 17 | C2≀S4 | 1+53T+15496T2+576511T3+103902926T4+576511p3T5+15496p6T6+53p9T7+p12T8 |
| 19 | C2≀S4 | 1+29T+6638T2+258553T3+53805642T4+258553p3T5+6638p6T6+29p9T7+p12T8 |
| 23 | C2≀S4 | 1−295T+54367T2−7704248T3+887991584T4−7704248p3T5+54367p6T6−295p9T7+p12T8 |
| 29 | C2≀S4 | 1+129T+83933T2+9167698T3+2922882282T4+9167698p3T5+83933p6T6+129p9T7+p12T8 |
| 31 | C2≀S4 | 1−114T+80728T2−5927082T3+3079701134T4−5927082p3T5+80728p6T6−114p9T7+p12T8 |
| 37 | C2≀S4 | 1+403T+202319T2+46625606T3+14127493020T4+46625606p3T5+202319p6T6+403p9T7+p12T8 |
| 41 | C2≀S4 | 1+671T+399120T2+139154877T3+44432315734T4+139154877p3T5+399120p6T6+671p9T7+p12T8 |
| 43 | C2≀S4 | 1−411T+213439T2−60368868T3+23440158536T4−60368868p3T5+213439p6T6−411p9T7+p12T8 |
| 47 | C2≀S4 | 1+8T+160820T2−20939512T3+12301264614T4−20939512p3T5+160820p6T6+8p9T7+p12T8 |
| 53 | C2≀S4 | 1−90T+133752T2+19514962T3+19633648894T4+19514962p3T5+133752p6T6−90p9T7+p12T8 |
| 59 | C2≀S4 | 1+1018T+1189512T2+683319466T3+407279566174T4+683319466p3T5+1189512p6T6+1018p9T7+p12T8 |
| 61 | C2≀S4 | 1−50T+722188T2−51306118T3+225994176278T4−51306118p3T5+722188p6T6−50p9T7+p12T8 |
| 67 | C2≀S4 | 1+424T+917806T2+255868016T3+369880128047T4+255868016p3T5+917806p6T6+424p9T7+p12T8 |
| 71 | C2≀S4 | 1+215T+982021T2+275350320T3+449702043326T4+275350320p3T5+982021p6T6+215p9T7+p12T8 |
| 73 | C2≀S4 | 1−1207T+1581740T2−1193773973T3+948033109470T4−1193773973p3T5+1581740p6T6−1207p9T7+p12T8 |
| 79 | C2≀S4 | 1+951T+1431057T2+665911232T3+746318767554T4+665911232p3T5+1431057p6T6+951p9T7+p12T8 |
| 83 | C2≀S4 | 1+3035T+5488190T2+6563721455T3+5816729995458T4+6563721455p3T5+5488190p6T6+3035p9T7+p12T8 |
| 89 | C2≀S4 | 1+2819T+5531118T2+7041312813T3+6955665197242T4+7041312813p3T5+5531118p6T6+2819p9T7+p12T8 |
| 97 | C2≀S4 | 1−1100T+625060T2+1205286580T3−1327506690522T4+1205286580p3T5+625060p6T6−1100p9T7+p12T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.83522105699567632849114607417, −6.66547057210302884916346173312, −6.08445837208276198508666334141, −6.07130404046609051719066019444, −5.87399555005394543990128888500, −5.70383852807925999194601332644, −5.27513364180265903058296641855, −5.21781657259045096229567898352, −5.21095000228852869295322055771, −4.66043961227287916805022355410, −4.63451614880122821168500082308, −4.42059033056585720743687983892, −4.39137653105078944536123181915, −3.77033592552118968377450768158, −3.43658528816878639627630275759, −3.42287669883448047211874415962, −3.06682260160857588527523355090, −2.93597061187632528117779620375, −2.73288405632729858763018071585, −2.62286434407456940370808181808, −2.17134385187422557534768265302, −1.85440271616855420699768383103, −1.83034871472492507952397496032, −1.03205557113987557879408677893, −1.01705184489022442378719325199, 0, 0, 0, 0,
1.01705184489022442378719325199, 1.03205557113987557879408677893, 1.83034871472492507952397496032, 1.85440271616855420699768383103, 2.17134385187422557534768265302, 2.62286434407456940370808181808, 2.73288405632729858763018071585, 2.93597061187632528117779620375, 3.06682260160857588527523355090, 3.42287669883448047211874415962, 3.43658528816878639627630275759, 3.77033592552118968377450768158, 4.39137653105078944536123181915, 4.42059033056585720743687983892, 4.63451614880122821168500082308, 4.66043961227287916805022355410, 5.21095000228852869295322055771, 5.21781657259045096229567898352, 5.27513364180265903058296641855, 5.70383852807925999194601332644, 5.87399555005394543990128888500, 6.07130404046609051719066019444, 6.08445837208276198508666334141, 6.66547057210302884916346173312, 6.83522105699567632849114607417