L(s) = 1 | + 3·5-s − 4·7-s − 8·11-s − 2·13-s − 2·17-s + 8·19-s + 6·25-s + 3·29-s + 4·31-s − 12·35-s − 10·37-s − 10·41-s + 14·43-s + 10·47-s − 49-s − 10·53-s − 24·55-s − 8·59-s − 22·61-s − 6·65-s − 12·67-s + 8·71-s − 2·73-s + 32·77-s − 12·83-s − 6·85-s − 18·89-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 1.51·7-s − 2.41·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 6/5·25-s + 0.557·29-s + 0.718·31-s − 2.02·35-s − 1.64·37-s − 1.56·41-s + 2.13·43-s + 1.45·47-s − 1/7·49-s − 1.37·53-s − 3.23·55-s − 1.04·59-s − 2.81·61-s − 0.744·65-s − 1.46·67-s + 0.949·71-s − 0.234·73-s + 3.64·77-s − 1.31·83-s − 0.650·85-s − 1.90·89-s + ⋯ |
Λ(s)=(=((26⋅36⋅53⋅293)s/2ΓC(s)3L(s)−Λ(2−s)
Λ(s)=(=((26⋅36⋅53⋅293)s/2ΓC(s+1/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
26⋅36⋅53⋅293
|
Sign: |
−1
|
Analytic conductor: |
72417.3 |
Root analytic conductor: |
6.45615 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 26⋅36⋅53⋅293, ( :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 |
| 3 | | 1 |
| 5 | C1 | (1−T)3 |
| 29 | C1 | (1−T)3 |
good | 7 | S4×C2 | 1+4T+17T2+52T3+17pT4+4p2T5+p3T6 |
| 11 | S4×C2 | 1+8T+45T2+164T3+45pT4+8p2T5+p3T6 |
| 13 | S4×C2 | 1+2T+19T2+28T3+19pT4+2p2T5+p3T6 |
| 17 | S4×C2 | 1+2T+15T2−40T3+15pT4+2p2T5+p3T6 |
| 19 | S4×C2 | 1−8T+41T2−140T3+41pT4−8p2T5+p3T6 |
| 23 | S4×C2 | 1+45T2+36T3+45pT4+p3T6 |
| 31 | S4×C2 | 1−4T+61T2−284T3+61pT4−4p2T5+p3T6 |
| 37 | S4×C2 | 1+10T+59T2+232T3+59pT4+10p2T5+p3T6 |
| 41 | S4×C2 | 1+10T+135T2+796T3+135pT4+10p2T5+p3T6 |
| 43 | D6 | 1−14T+113T2−656T3+113pT4−14p2T5+p3T6 |
| 47 | S4×C2 | 1−10T+165T2−952T3+165pT4−10p2T5+p3T6 |
| 53 | S4×C2 | 1+10T+147T2+988T3+147pT4+10p2T5+p3T6 |
| 59 | S4×C2 | 1+8T+9T2−544T3+9pT4+8p2T5+p3T6 |
| 61 | D6 | 1+22T+299T2+2788T3+299pT4+22p2T5+p3T6 |
| 67 | S4×C2 | 1+12T+225T2+1540T3+225pT4+12p2T5+p3T6 |
| 71 | S4×C2 | 1−8T+189T2−992T3+189pT4−8p2T5+p3T6 |
| 73 | S4×C2 | 1+2T+139T2−32T3+139pT4+2p2T5+p3T6 |
| 79 | S4×C2 | 1+129T2+244T3+129pT4+p3T6 |
| 83 | S4×C2 | 1+12T+213T2+1884T3+213pT4+12p2T5+p3T6 |
| 89 | S4×C2 | 1+18T+279T2+3132T3+279pT4+18p2T5+p3T6 |
| 97 | S4×C2 | 1+38T+763T2+9280T3+763pT4+38p2T5+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
−7.59856827358761333804860977260, −7.35118778127331763399440308823, −7.14200907682236930242847764227, −6.92687192791166052311481522380, −6.56033229582637809809321863363, −6.34785680134332272186749732621, −6.26439082449963576541415048603, −5.95375970594210402773554882291, −5.51592119291581419662411843173, −5.42716492047027328077659318231, −5.26283226594296558626517902941, −5.01405023583652197521142122963, −4.92846684637236090465011592352, −4.34484038441904270340928245055, −4.07560427281879546221822867948, −3.99316201191640441343857693944, −3.31387031175211363420609494670, −3.05966648970538958709406427185, −2.95193618032618234049668460411, −2.68416228837162531807896907641, −2.54456662855585889951573625539, −2.34865965233774095571105588383, −1.44087116678613878655112741439, −1.42096228406856806185233043419, −1.27692092150067006038620236935, 0, 0, 0,
1.27692092150067006038620236935, 1.42096228406856806185233043419, 1.44087116678613878655112741439, 2.34865965233774095571105588383, 2.54456662855585889951573625539, 2.68416228837162531807896907641, 2.95193618032618234049668460411, 3.05966648970538958709406427185, 3.31387031175211363420609494670, 3.99316201191640441343857693944, 4.07560427281879546221822867948, 4.34484038441904270340928245055, 4.92846684637236090465011592352, 5.01405023583652197521142122963, 5.26283226594296558626517902941, 5.42716492047027328077659318231, 5.51592119291581419662411843173, 5.95375970594210402773554882291, 6.26439082449963576541415048603, 6.34785680134332272186749732621, 6.56033229582637809809321863363, 6.92687192791166052311481522380, 7.14200907682236930242847764227, 7.35118778127331763399440308823, 7.59856827358761333804860977260