L(s) = 1 | − 2·2-s + 27·3-s − 67·4-s − 54·6-s + 68·7-s + 166·8-s + 486·9-s + 363·11-s − 1.80e3·12-s − 290·13-s − 136·14-s + 2.37e3·16-s − 434·17-s − 972·18-s − 2.85e3·19-s + 1.83e3·21-s − 726·22-s + 640·23-s + 4.48e3·24-s + 580·26-s + 7.29e3·27-s − 4.55e3·28-s − 4.53e3·29-s − 1.49e4·31-s − 6.66e3·32-s + 9.80e3·33-s + 868·34-s + ⋯ |
L(s) = 1 | − 0.353·2-s + 1.73·3-s − 2.09·4-s − 0.612·6-s + 0.524·7-s + 0.917·8-s + 2·9-s + 0.904·11-s − 3.62·12-s − 0.475·13-s − 0.185·14-s + 2.31·16-s − 0.364·17-s − 0.707·18-s − 1.81·19-s + 0.908·21-s − 0.319·22-s + 0.252·23-s + 1.58·24-s + 0.168·26-s + 1.92·27-s − 1.09·28-s − 1.00·29-s − 2.79·31-s − 1.15·32-s + 1.56·33-s + 0.128·34-s + ⋯ |
Λ(s)=(=((33⋅56⋅113)s/2ΓC(s)3L(s)−Λ(6−s)
Λ(s)=(=((33⋅56⋅113)s/2ΓC(s+5/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
33⋅56⋅113
|
Sign: |
−1
|
Analytic conductor: |
2.31655×106 |
Root analytic conductor: |
11.5028 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 33⋅56⋅113, ( :5/2,5/2,5/2), −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 | 3 | C1 | (1−p2T)3 |
| 5 | | 1 |
| 11 | C1 | (1−p2T)3 |
good | 2 | S4×C2 | 1+pT+71T2+55pT3+71p5T4+p11T5+p15T6 |
| 7 | S4×C2 | 1−68T+34869T2−2533560T3+34869p5T4−68p10T5+p15T6 |
| 13 | S4×C2 | 1+290T+658819T2+83286348T3+658819p5T4+290p10T5+p15T6 |
| 17 | S4×C2 | 1+434T+58991T2−1314616612T3+58991p5T4+434p10T5+p15T6 |
| 19 | S4×C2 | 1+2856T+8753113T2+14281181168T3+8753113p5T4+2856p10T5+p15T6 |
| 23 | S4×C2 | 1−640T+6700261T2+7538830592T3+6700261p5T4−640p10T5+p15T6 |
| 29 | S4×C2 | 1+4538T+35092643T2+141745639868T3+35092643p5T4+4538p10T5+p15T6 |
| 31 | S4×C2 | 1+14968T+160071261T2+978687787280T3+160071261p5T4+14968p10T5+p15T6 |
| 37 | S4×C2 | 1−6190T+113089771T2−886491849396T3+113089771p5T4−6190p10T5+p15T6 |
| 41 | S4×C2 | 1+8926T+272145047T2+2187570068644T3+272145047p5T4+8926p10T5+p15T6 |
| 43 | S4×C2 | 1−33592T+693464257T2−9837617686992T3+693464257p5T4−33592p10T5+p15T6 |
| 47 | S4×C2 | 1−24640T+788101261T2−10622124840832T3+788101261p5T4−24640p10T5+p15T6 |
| 53 | S4×C2 | 1−22934T+1099334651T2−14788031800868T3+1099334651p5T4−22934p10T5+p15T6 |
| 59 | S4×C2 | 1+13756T−129165359T2−1129007325848T3−129165359p5T4+13756p10T5+p15T6 |
| 61 | S4×C2 | 1−24602T+2638007155T2−41514451599900T3+2638007155p5T4−24602p10T5+p15T6 |
| 67 | S4×C2 | 1+16868T+3843672649T2+43721067889560T3+3843672649p5T4+16868p10T5+p15T6 |
| 71 | S4×C2 | 1−4856T+1951490165T2+16632390591088T3+1951490165p5T4−4856p10T5+p15T6 |
| 73 | S4×C2 | 1+1910T+5875530055T2+8082237219348T3+5875530055p5T4+1910p10T5+p15T6 |
| 79 | S4×C2 | 1+36844T+4302241965T2+155969878390184T3+4302241965p5T4+36844p10T5+p15T6 |
| 83 | S4×C2 | 1−48796T+9666032953T2−314032777245928T3+9666032953p5T4−48796p10T5+p15T6 |
| 89 | S4×C2 | 1+188978T+25306287767T2+2094440123430812T3+25306287767p5T4+188978p10T5+p15T6 |
| 97 | S4×C2 | 1+247526T+41254986031T2+4400049090000852T3+41254986031p5T4+247526p10T5+p15T6 |
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L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.867436054936391435373700549880, −8.428959922781021647355991791509, −8.336691385767081788545429421265, −8.323082667941505028636600858187, −7.52952586048962628361150951615, −7.47924671998887287775658630026, −7.42310795034946919358928784140, −6.91401248555190957101366837079, −6.46062820198606362844246076825, −6.26873064167609406957357295898, −5.52730126559955464673100110921, −5.48504101154572855845718291019, −5.23806053301703231394592305812, −4.48601481583481356968311315583, −4.48322476342371146097257545787, −4.20850074338068593651839012484, −3.80315682864487268670107460561, −3.77849809297233710936134131110, −3.41723391003108686861821456561, −2.55672510736687406640517922920, −2.49975657489757273391537176730, −2.19454320605859059652973046126, −1.55542013325244813116500576175, −1.16379482382485122669916559587, −1.14705849442005581393992145520, 0, 0, 0,
1.14705849442005581393992145520, 1.16379482382485122669916559587, 1.55542013325244813116500576175, 2.19454320605859059652973046126, 2.49975657489757273391537176730, 2.55672510736687406640517922920, 3.41723391003108686861821456561, 3.77849809297233710936134131110, 3.80315682864487268670107460561, 4.20850074338068593651839012484, 4.48322476342371146097257545787, 4.48601481583481356968311315583, 5.23806053301703231394592305812, 5.48504101154572855845718291019, 5.52730126559955464673100110921, 6.26873064167609406957357295898, 6.46062820198606362844246076825, 6.91401248555190957101366837079, 7.42310795034946919358928784140, 7.47924671998887287775658630026, 7.52952586048962628361150951615, 8.323082667941505028636600858187, 8.336691385767081788545429421265, 8.428959922781021647355991791509, 8.867436054936391435373700549880