L(s) = 1 | + (−3.93 − 0.723i)2-s + (14.9 + 5.69i)4-s − 10.6·5-s + 17.1·7-s + (−54.7 − 33.2i)8-s + (41.9 + 7.72i)10-s − 55.4·11-s + 95.6i·13-s + (−67.3 − 12.3i)14-s + (191. + 170. i)16-s + 365. i·17-s + 499. i·19-s + (−159. − 60.7i)20-s + (218. + 40.1i)22-s + 758. i·23-s + ⋯ |
L(s) = 1 | + (−0.983 − 0.180i)2-s + (0.934 + 0.355i)4-s − 0.426·5-s + 0.349·7-s + (−0.854 − 0.519i)8-s + (0.419 + 0.0772i)10-s − 0.458·11-s + 0.565i·13-s + (−0.343 − 0.0632i)14-s + (0.746 + 0.665i)16-s + 1.26i·17-s + 1.38i·19-s + (−0.398 − 0.151i)20-s + (0.450 + 0.0829i)22-s + 1.43i·23-s + ⋯ |
Λ(s)=(=(72s/2ΓC(s)L(s)(−0.0695−0.997i)Λ(5−s)
Λ(s)=(=(72s/2ΓC(s+2)L(s)(−0.0695−0.997i)Λ(1−s)
Degree: |
2 |
Conductor: |
72
= 23⋅32
|
Sign: |
−0.0695−0.997i
|
Analytic conductor: |
7.44263 |
Root analytic conductor: |
2.72811 |
Motivic weight: |
4 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ72(53,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 72, ( :2), −0.0695−0.997i)
|
Particular Values
L(25) |
≈ |
0.454782+0.487615i |
L(21) |
≈ |
0.454782+0.487615i |
L(3) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(3.93+0.723i)T |
| 3 | 1 |
good | 5 | 1+10.6T+625T2 |
| 7 | 1−17.1T+2.40e3T2 |
| 11 | 1+55.4T+1.46e4T2 |
| 13 | 1−95.6iT−2.85e4T2 |
| 17 | 1−365.iT−8.35e4T2 |
| 19 | 1−499.iT−1.30e5T2 |
| 23 | 1−758.iT−2.79e5T2 |
| 29 | 1−958.T+7.07e5T2 |
| 31 | 1+830.T+9.23e5T2 |
| 37 | 1+2.03e3iT−1.87e6T2 |
| 41 | 1−319.iT−2.82e6T2 |
| 43 | 1−1.47e3iT−3.41e6T2 |
| 47 | 1−847.iT−4.87e6T2 |
| 53 | 1−3.83e3T+7.89e6T2 |
| 59 | 1+3.44e3T+1.21e7T2 |
| 61 | 1+4.80e3iT−1.38e7T2 |
| 67 | 1−8.02e3iT−2.01e7T2 |
| 71 | 1+2.68e3iT−2.54e7T2 |
| 73 | 1−4.19e3T+2.83e7T2 |
| 79 | 1−9.14e3T+3.89e7T2 |
| 83 | 1+1.15e4T+4.74e7T2 |
| 89 | 1+1.02e4iT−6.27e7T2 |
| 97 | 1−5.59e3T+8.85e7T2 |
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
−14.37586632834540835056264408308, −12.75961472742197557825730296330, −11.74573769206442053503265411724, −10.76344211077597439582563417828, −9.667385487573285565511891208718, −8.331962714977032514705803637654, −7.51934786691763095332372552359, −5.94227223864484288835409365791, −3.74356556023582714697307525459, −1.72247726357328550555351166596,
0.47455251476122259676435173044, 2.67744118117683287128136618820, 5.04204205382780055414760194579, 6.75136019193750077726744367491, 7.85845322355295652362538349274, 8.884101086370450508316051111833, 10.18496224987269251485775657036, 11.19837409121303633188234729550, 12.17628267400534626346693459660, 13.72092545311905054318224240088