L(s) = 1 | − 7·8-s − 69·23-s + 75·25-s − 38·27-s + 147·49-s + 78·59-s − 15·64-s − 498·101-s + 363·121-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 + 483·184-s + 191-s + 193-s + 197-s + 199-s − 525·200-s + ⋯ |
L(s) = 1 | − 7/8·8-s − 3·23-s + 3·25-s − 1.40·27-s + 3·49-s + 1.32·59-s − 0.234·64-s − 4.93·101-s + 3·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 21/8·184-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s − 2.62·200-s + ⋯ |
Λ(s)=(=(12167s/2ΓC(s)3L(s)Λ(3−s)
Λ(s)=(=(12167s/2ΓC(s+1)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
12167
= 233
|
Sign: |
1
|
Analytic conductor: |
0.246143 |
Root analytic conductor: |
0.791646 |
Motivic weight: |
2 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
induced by χ23(22,⋅)
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 12167, ( :1,1,1), 1)
|
Particular Values
L(23) |
≈ |
0.7038092951 |
L(21) |
≈ |
0.7038092951 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 23 | C1 | (1+pT)3 |
good | 2 | D6 | 1+7T3+p6T6 |
| 3 | D6 | 1+38T3+p6T6 |
| 5 | C1×C1 | (1−pT)3(1+pT)3 |
| 7 | C1×C1 | (1−pT)3(1+pT)3 |
| 11 | C1×C1 | (1−pT)3(1+pT)3 |
| 13 | D6 | 1−1082T3+p6T6 |
| 17 | C1×C1 | (1−pT)3(1+pT)3 |
| 19 | C1×C1 | (1−pT)3(1+pT)3 |
| 29 | D6 | 1−30746T3+p6T6 |
| 31 | D6 | 1−58754T3+p6T6 |
| 37 | C1×C1 | (1−pT)3(1+pT)3 |
| 41 | D6 | 1−43634T3+p6T6 |
| 43 | C1×C1 | (1−pT)3(1+pT)3 |
| 47 | D6 | 1+205342T3+p6T6 |
| 53 | C1×C1 | (1−pT)3(1+pT)3 |
| 59 | C2 | (1−26T+p2T2)3 |
| 61 | C1×C1 | (1−pT)3(1+pT)3 |
| 67 | C1×C1 | (1−pT)3(1+pT)3 |
| 71 | D6 | 1−667154T3+p6T6 |
| 73 | D6 | 1−725042T3+p6T6 |
| 79 | C1×C1 | (1−pT)3(1+pT)3 |
| 83 | C1×C1 | (1−pT)3(1+pT)3 |
| 89 | C1×C1 | (1−pT)3(1+pT)3 |
| 97 | C1×C1 | (1−pT)3(1+pT)3 |
show more | | |
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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
−16.25266090259699302544844085089, −15.38379949405769162888300515781, −15.25045957856847525792594451566, −14.82823558841975961978977417582, −14.30448934002337246511345373331, −13.87709638210915675759205960405, −13.54348074276356119184964203974, −12.88326648893801043752994094227, −12.34454102311100891116481500791, −12.18652506141296656813498136593, −11.66311862169944965457286273302, −11.12595221769137640392299636957, −10.35740039286693641585425892371, −10.33664492004809423462870333956, −9.319600292021350002126565316276, −9.255683549229959107838401341587, −8.254627656874356514129115296000, −8.249320660797426125572295542019, −7.15878206375237697440477144189, −6.81039803586467718459107985552, −5.79533420479259215020430426704, −5.67469663404156687062790353124, −4.46007887250445186823029671800, −3.69917499700457382202237665871, −2.50316094689835549482408964468,
2.50316094689835549482408964468, 3.69917499700457382202237665871, 4.46007887250445186823029671800, 5.67469663404156687062790353124, 5.79533420479259215020430426704, 6.81039803586467718459107985552, 7.15878206375237697440477144189, 8.249320660797426125572295542019, 8.254627656874356514129115296000, 9.255683549229959107838401341587, 9.319600292021350002126565316276, 10.33664492004809423462870333956, 10.35740039286693641585425892371, 11.12595221769137640392299636957, 11.66311862169944965457286273302, 12.18652506141296656813498136593, 12.34454102311100891116481500791, 12.88326648893801043752994094227, 13.54348074276356119184964203974, 13.87709638210915675759205960405, 14.30448934002337246511345373331, 14.82823558841975961978977417582, 15.25045957856847525792594451566, 15.38379949405769162888300515781, 16.25266090259699302544844085089