L(s) = 1 | + (5.53 + 9.58i)2-s + (136. + 32.8i)3-s + (194. − 337. i)4-s + (312.5 − 541. i)5-s + (439. + 1.48e3i)6-s + (−2.33e3 − 4.05e3i)7-s + 9.97e3·8-s + (1.75e4 + 8.97e3i)9-s + 6.91e3·10-s + (−3.42e4 − 5.93e4i)11-s + (3.76e4 − 3.95e4i)12-s + (2.05e4 − 3.55e4i)13-s + (2.58e4 − 4.48e4i)14-s + (6.04e4 − 6.35e4i)15-s + (−4.44e4 − 7.69e4i)16-s − 3.96e5·17-s + ⋯ |
L(s) = 1 | + (0.244 + 0.423i)2-s + (0.972 + 0.234i)3-s + (0.380 − 0.658i)4-s + (0.223 − 0.387i)5-s + (0.138 + 0.469i)6-s + (−0.368 − 0.637i)7-s + 0.861·8-s + (0.890 + 0.455i)9-s + 0.218·10-s + (−0.705 − 1.22i)11-s + (0.524 − 0.551i)12-s + (0.199 − 0.344i)13-s + (0.180 − 0.312i)14-s + (0.308 − 0.324i)15-s + (−0.169 − 0.293i)16-s − 1.15·17-s + ⋯ |
Λ(s)=(=(45s/2ΓC(s)L(s)(0.680+0.732i)Λ(10−s)
Λ(s)=(=(45s/2ΓC(s+9/2)L(s)(0.680+0.732i)Λ(1−s)
Degree: |
2 |
Conductor: |
45
= 32⋅5
|
Sign: |
0.680+0.732i
|
Analytic conductor: |
23.1766 |
Root analytic conductor: |
4.81420 |
Motivic weight: |
9 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ45(31,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 45, ( :9/2), 0.680+0.732i)
|
Particular Values
L(5) |
≈ |
3.06477−1.33641i |
L(21) |
≈ |
3.06477−1.33641i |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−136.−32.8i)T |
| 5 | 1+(−312.5+541.i)T |
good | 2 | 1+(−5.53−9.58i)T+(−256+443.i)T2 |
| 7 | 1+(2.33e3+4.05e3i)T+(−2.01e7+3.49e7i)T2 |
| 11 | 1+(3.42e4+5.93e4i)T+(−1.17e9+2.04e9i)T2 |
| 13 | 1+(−2.05e4+3.55e4i)T+(−5.30e9−9.18e9i)T2 |
| 17 | 1+3.96e5T+1.18e11T2 |
| 19 | 1−1.49e5T+3.22e11T2 |
| 23 | 1+(−3.97e4+6.89e4i)T+(−9.00e11−1.55e12i)T2 |
| 29 | 1+(−1.92e6−3.32e6i)T+(−7.25e12+1.25e13i)T2 |
| 31 | 1+(−2.21e6+3.83e6i)T+(−1.32e13−2.28e13i)T2 |
| 37 | 1−1.52e7T+1.29e14T2 |
| 41 | 1+(−2.61e5+4.53e5i)T+(−1.63e14−2.83e14i)T2 |
| 43 | 1+(−1.36e7−2.37e7i)T+(−2.51e14+4.35e14i)T2 |
| 47 | 1+(−6.19e6−1.07e7i)T+(−5.59e14+9.69e14i)T2 |
| 53 | 1−7.97e7T+3.29e15T2 |
| 59 | 1+(7.60e7−1.31e8i)T+(−4.33e15−7.50e15i)T2 |
| 61 | 1+(9.76e6+1.69e7i)T+(−5.84e15+1.01e16i)T2 |
| 67 | 1+(1.20e8−2.09e8i)T+(−1.36e16−2.35e16i)T2 |
| 71 | 1+2.11e8T+4.58e16T2 |
| 73 | 1+1.03e8T+5.88e16T2 |
| 79 | 1+(1.51e8+2.62e8i)T+(−5.99e16+1.03e17i)T2 |
| 83 | 1+(−1.93e8−3.35e8i)T+(−9.34e16+1.61e17i)T2 |
| 89 | 1−2.21e8T+3.50e17T2 |
| 97 | 1+(8.12e8+1.40e9i)T+(−3.80e17+6.58e17i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.65347681375599167665827024709, −13.23677043655755224425785638655, −10.99938307602546327757977281670, −10.07013173881042954977649169001, −8.714178443269833852433262398588, −7.41633423176483416714755952009, −5.96310505929676114308696826758, −4.43900655581747950065342322635, −2.71386069636870607375979476495, −0.952943584969128876169701435465,
2.04315444526990253482220733425, 2.78929645187024872930570632401, 4.32798442981806363036616932483, 6.67457074849458653695284826580, 7.75148276596572506849012242261, 9.109620732815825886001787034105, 10.39955524951107684191983179146, 11.94128842720801025428434810517, 12.89007801078234502520265734634, 13.73480539826958060003398577875