L(s) = 1 | + 2·3-s + 3·5-s − 9-s − 5·11-s − 2·13-s + 6·15-s − 10·17-s − 3·19-s − 17·23-s − 25-s − 6·27-s + 2·29-s − 6·31-s − 10·33-s + 14·37-s − 4·39-s − 4·41-s − 17·43-s − 3·45-s + 9·47-s − 20·51-s − 22·53-s − 15·55-s − 6·57-s − 6·59-s − 61-s − 6·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.34·5-s − 1/3·9-s − 1.50·11-s − 0.554·13-s + 1.54·15-s − 2.42·17-s − 0.688·19-s − 3.54·23-s − 1/5·25-s − 1.15·27-s + 0.371·29-s − 1.07·31-s − 1.74·33-s + 2.30·37-s − 0.640·39-s − 0.624·41-s − 2.59·43-s − 0.447·45-s + 1.31·47-s − 2.80·51-s − 3.02·53-s − 2.02·55-s − 0.794·57-s − 0.781·59-s − 0.128·61-s − 0.744·65-s + ⋯ |
Λ(s)=(=((26⋅76⋅193)s/2ΓC(s)3L(s)−Λ(2−s)
Λ(s)=(=((26⋅76⋅193)s/2ΓC(s+1/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
26⋅76⋅193
|
Sign: |
−1
|
Analytic conductor: |
26294.2 |
Root analytic conductor: |
5.45309 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 26⋅76⋅193, ( :1/2,1/2,1/2), −1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 7 | | 1 |
| 19 | C1 | (1+T)3 |
good | 3 | S4×C2 | 1−2T+5T2−2pT3+5pT4−2p2T5+p3T6 |
| 5 | S4×C2 | 1−3T+2pT2−19T3+2p2T4−3p2T5+p3T6 |
| 11 | S4×C2 | 1+5T+34T2+109T3+34pT4+5p2T5+p3T6 |
| 13 | S4×C2 | 1+2T+33T2+54T3+33pT4+2p2T5+p3T6 |
| 17 | S4×C2 | 1+10T+67T2+304T3+67pT4+10p2T5+p3T6 |
| 23 | S4×C2 | 1+17T+160T2+931T3+160pT4+17p2T5+p3T6 |
| 29 | S4×C2 | 1−2T+pT2−134T3+p2T4−2p2T5+p3T6 |
| 31 | S4×C2 | 1+6T+57T2+358T3+57pT4+6p2T5+p3T6 |
| 37 | S4×C2 | 1−14T+169T2−30pT3+169pT4−14p2T5+p3T6 |
| 41 | S4×C2 | 1+4T+51T2+472T3+51pT4+4p2T5+p3T6 |
| 43 | S4×C2 | 1+17T+220T2+1611T3+220pT4+17p2T5+p3T6 |
| 47 | S4×C2 | 1−9T+124T2−735T3+124pT4−9p2T5+p3T6 |
| 53 | S4×C2 | 1+22T+225T2+1706T3+225pT4+22p2T5+p3T6 |
| 59 | C2 | (1+2T+pT2)3 |
| 61 | S4×C2 | 1+T+126T2+261T3+126pT4+p2T5+p3T6 |
| 67 | S4×C2 | 1−2T+133T2−100T3+133pT4−2p2T5+p3T6 |
| 71 | S4×C2 | 1+18T+277T2+2514T3+277pT4+18p2T5+p3T6 |
| 73 | S4×C2 | 1−21T+334T2−3217T3+334pT4−21p2T5+p3T6 |
| 79 | S4×C2 | 1−8T+253T2−1266T3+253pT4−8p2T5+p3T6 |
| 83 | S4×C2 | 1+27T+426T2+4439T3+426pT4+27p2T5+p3T6 |
| 89 | S4×C2 | 1−30T+501T2−5502T3+501pT4−30p2T5+p3T6 |
| 97 | S4×C2 | 1+14T+119T2+1252T3+119pT4+14p2T5+p3T6 |
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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
−8.099669021231709156958361399414, −7.62795009869192896599743433183, −7.60230793182888894231565949875, −7.30681525802612281388261918008, −6.96253551361268373923940179101, −6.43222414792494547575130884634, −6.40064348689333503799455181735, −6.06538119631104881468455499311, −6.03762561085899826738023943401, −5.86789917482284339606472000489, −5.40641295031324779674898730289, −5.05848951606485923657501294482, −4.83534027009847245266282219790, −4.54651051521495809708834312671, −4.45480908886158491450415062604, −3.94133713281796765685396976345, −3.59944191780398294406207318495, −3.54144210120259368279489682961, −3.00543222224667717052983590157, −2.51580343142356178859126788996, −2.43915757792854877372342510325, −2.35042024607911817423970034714, −1.93839213548530716833934848952, −1.80714947930969138668345858766, −1.33680118005179253749863889804, 0, 0, 0,
1.33680118005179253749863889804, 1.80714947930969138668345858766, 1.93839213548530716833934848952, 2.35042024607911817423970034714, 2.43915757792854877372342510325, 2.51580343142356178859126788996, 3.00543222224667717052983590157, 3.54144210120259368279489682961, 3.59944191780398294406207318495, 3.94133713281796765685396976345, 4.45480908886158491450415062604, 4.54651051521495809708834312671, 4.83534027009847245266282219790, 5.05848951606485923657501294482, 5.40641295031324779674898730289, 5.86789917482284339606472000489, 6.03762561085899826738023943401, 6.06538119631104881468455499311, 6.40064348689333503799455181735, 6.43222414792494547575130884634, 6.96253551361268373923940179101, 7.30681525802612281388261918008, 7.60230793182888894231565949875, 7.62795009869192896599743433183, 8.099669021231709156958361399414