L(s) = 1 | − 1.95e3·8-s + 8.39e5·23-s + 1.17e6·25-s − 1.06e6·27-s + 1.72e7·49-s + 4.58e7·59-s − 1.29e7·64-s − 4.70e8·101-s + 6.43e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 1.63e9·184-s + 191-s + 193-s + 197-s + 199-s − 2.28e9·200-s + ⋯ |
L(s) = 1 | − 0.476·8-s + 3·23-s + 3·25-s − 1.99·27-s + 3·49-s + 3.78·59-s − 0.773·64-s − 4.52·101-s + 3·121-s − 1.42·184-s − 1.42·200-s + 0.952·216-s + ⋯ |
Λ(s)=(=(12167s/2ΓC(s)3L(s)Λ(9−s)
Λ(s)=(=(12167s/2ΓC(s+4)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
12167
= 233
|
Sign: |
1
|
Analytic conductor: |
822.580 |
Root analytic conductor: |
3.06099 |
Motivic weight: |
8 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
induced by χ23(22,⋅)
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 12167, ( :4,4,4), 1)
|
Particular Values
L(29) |
≈ |
3.101080880 |
L(21) |
≈ |
3.101080880 |
L(5) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 23 | C1 | (1−p4T)3 |
good | 2 | D6 | 1+1951T3+p24T6 |
| 3 | D6 | 1+1062686T3+p24T6 |
| 5 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 7 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 11 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 13 | D6 | 1−25363320370274T3+p24T6 |
| 17 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 19 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 29 | D6 | 1+647932355939762206T3+p24T6 |
| 31 | D6 | 1−1237087799571624194T3+p24T6 |
| 37 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 41 | D6 | 1−12576527614080568514T3+p24T6 |
| 43 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 47 | D6 | 1−19⋯54T3+p24T6 |
| 53 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 59 | C2 | (1−15279074T+p8T2)3 |
| 61 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 67 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 71 | D6 | 1−28⋯94T3+p24T6 |
| 73 | D6 | 1−39⋯54T3+p24T6 |
| 79 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 83 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 89 | C1×C1 | (1−p4T)3(1+p4T)3 |
| 97 | C1×C1 | (1−p4T)3(1+p4T)3 |
show more | | |
show less | | |
L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−14.60505482600838178979882243183, −13.57754776366493196214640179035, −13.57358699137694629103351250428, −12.94340974939646264484421352028, −12.58994809827745075575670473695, −12.20573812798288630229092135669, −11.48934113174410361454083235145, −11.13020128337881722518285137089, −10.78407804078334704513648951630, −10.26071600882990030330086952673, −9.569195258661682746768575480396, −9.047364399505374511119356771709, −8.785528370880907108738112345945, −8.312193720324238660650046295629, −7.22331694672823028526268534820, −7.12821196931798717727353152062, −6.61290661913085672954290949270, −5.48365710965517680196720795192, −5.41807275592759405744549257376, −4.54563824817196368509283972522, −3.74241792710594257007497295027, −2.95312795013608813421629405734, −2.44985366376325530211392602373, −1.15438384073458975501791693746, −0.66124939014099614407429649788,
0.66124939014099614407429649788, 1.15438384073458975501791693746, 2.44985366376325530211392602373, 2.95312795013608813421629405734, 3.74241792710594257007497295027, 4.54563824817196368509283972522, 5.41807275592759405744549257376, 5.48365710965517680196720795192, 6.61290661913085672954290949270, 7.12821196931798717727353152062, 7.22331694672823028526268534820, 8.312193720324238660650046295629, 8.785528370880907108738112345945, 9.047364399505374511119356771709, 9.569195258661682746768575480396, 10.26071600882990030330086952673, 10.78407804078334704513648951630, 11.13020128337881722518285137089, 11.48934113174410361454083235145, 12.20573812798288630229092135669, 12.58994809827745075575670473695, 12.94340974939646264484421352028, 13.57358699137694629103351250428, 13.57754776366493196214640179035, 14.60505482600838178979882243183