L(s) = 1 | − 4·2-s − 3·3-s + 10·4-s + 12·6-s − 17·8-s − 19·9-s + 100·11-s − 30·12-s + 44·13-s − 10·16-s − 53·17-s + 76·18-s + 29·19-s − 400·22-s − 295·23-s + 51·24-s − 176·26-s − 54·27-s + 129·29-s − 114·31-s + 212·32-s − 300·33-s + 212·34-s − 190·36-s − 403·37-s − 116·38-s − 132·39-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 5/4·4-s + 0.816·6-s − 0.751·8-s − 0.703·9-s + 2.74·11-s − 0.721·12-s + 0.938·13-s − 0.156·16-s − 0.756·17-s + 0.995·18-s + 0.350·19-s − 3.87·22-s − 2.67·23-s + 0.433·24-s − 1.32·26-s − 0.384·27-s + 0.826·29-s − 0.660·31-s + 1.17·32-s − 1.58·33-s + 1.06·34-s − 0.879·36-s − 1.79·37-s − 0.495·38-s − 0.541·39-s + ⋯ |
Λ(s)=(=((58⋅78)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((58⋅78)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
58⋅78
|
Sign: |
1
|
Analytic conductor: |
2.72903×107 |
Root analytic conductor: |
8.50160 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 58⋅78, ( :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 | 5 | | 1 |
| 7 | | 1 |
good | 2 | C2≀S4 | 1+p2T+3pT2+T3+11pT4+p3T5+3p7T6+p11T7+p12T8 |
| 3 | C2≀S4 | 1+pT+28T2+65pT3+1150T4+65p4T5+28p6T6+p10T7+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
−6.90577747017518638876188848603, −6.86549271268285509763267693644, −6.73699440912720794576766934268, −6.31611469324170910101306684115, −6.10266817647913224128306225781, −5.94199711432685700350682523443, −5.88900146734510807040959223837, −5.59882092376569047550683865782, −5.47377215237084821810903600770, −4.90469260113944826905754662155, −4.74744492993242245056402390935, −4.43289551386949779488541864361, −4.17715055740484143193984492793, −3.93398383104174661985325003099, −3.88066246383256787242973696272, −3.67396005452955038873843986799, −3.13985392336095749798502308397, −2.90755977607247918896845824918, −2.73129773264815658185034573426, −2.38923472622279067456399828313, −1.76817491752824106549789722053, −1.64558723118750688615709789919, −1.45064870382712560056630914185, −1.35088420252162443431386266723, −1.06167432870786508679043778622, 0, 0, 0, 0,
1.06167432870786508679043778622, 1.35088420252162443431386266723, 1.45064870382712560056630914185, 1.64558723118750688615709789919, 1.76817491752824106549789722053, 2.38923472622279067456399828313, 2.73129773264815658185034573426, 2.90755977607247918896845824918, 3.13985392336095749798502308397, 3.67396005452955038873843986799, 3.88066246383256787242973696272, 3.93398383104174661985325003099, 4.17715055740484143193984492793, 4.43289551386949779488541864361, 4.74744492993242245056402390935, 4.90469260113944826905754662155, 5.47377215237084821810903600770, 5.59882092376569047550683865782, 5.88900146734510807040959223837, 5.94199711432685700350682523443, 6.10266817647913224128306225781, 6.31611469324170910101306684115, 6.73699440912720794576766934268, 6.86549271268285509763267693644, 6.90577747017518638876188848603