L(s) = 1 | + (−11.8 − 6.86i)2-s + (30.2 + 52.3i)4-s + (99.9 − 173. i)5-s + (−671. − 610. i)7-s + 927. i·8-s + (−2.37e3 + 1.37e3i)10-s + (4.94e3 − 2.85e3i)11-s − 1.05e4i·13-s + (3.78e3 + 1.18e4i)14-s + (1.02e4 − 1.77e4i)16-s + (−7.19e3 − 1.24e4i)17-s + (−3.28e4 − 1.89e4i)19-s + 1.20e4·20-s − 7.83e4·22-s + (4.61e4 + 2.66e4i)23-s + ⋯ |
L(s) = 1 | + (−1.05 − 0.606i)2-s + (0.236 + 0.408i)4-s + (0.357 − 0.619i)5-s + (−0.739 − 0.672i)7-s + 0.640i·8-s + (−0.751 + 0.433i)10-s + (1.11 − 0.646i)11-s − 1.33i·13-s + (0.369 + 1.15i)14-s + (0.624 − 1.08i)16-s + (−0.355 − 0.614i)17-s + (−1.09 − 0.633i)19-s + 0.337·20-s − 1.56·22-s + (0.790 + 0.456i)23-s + ⋯ |
Λ(s)=(=(63s/2ΓC(s)L(s)(−0.715−0.698i)Λ(8−s)
Λ(s)=(=(63s/2ΓC(s+7/2)L(s)(−0.715−0.698i)Λ(1−s)
Degree: |
2 |
Conductor: |
63
= 32⋅7
|
Sign: |
−0.715−0.698i
|
Analytic conductor: |
19.6802 |
Root analytic conductor: |
4.43624 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ63(26,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 63, ( :7/2), −0.715−0.698i)
|
Particular Values
L(4) |
≈ |
0.179940+0.442174i |
L(21) |
≈ |
0.179940+0.442174i |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(671.+610.i)T |
good | 2 | 1+(11.8+6.86i)T+(64+110.i)T2 |
| 5 | 1+(−99.9+173.i)T+(−3.90e4−6.76e4i)T2 |
| 11 | 1+(−4.94e3+2.85e3i)T+(9.74e6−1.68e7i)T2 |
| 13 | 1+1.05e4iT−6.27e7T2 |
| 17 | 1+(7.19e3+1.24e4i)T+(−2.05e8+3.55e8i)T2 |
| 19 | 1+(3.28e4+1.89e4i)T+(4.46e8+7.74e8i)T2 |
| 23 | 1+(−4.61e4−2.66e4i)T+(1.70e9+2.94e9i)T2 |
| 29 | 1−1.11e5iT−1.72e10T2 |
| 31 | 1+(1.34e5−7.75e4i)T+(1.37e10−2.38e10i)T2 |
| 37 | 1+(2.03e4−3.51e4i)T+(−4.74e10−8.22e10i)T2 |
| 41 | 1+6.93e5T+1.94e11T2 |
| 43 | 1+5.75e5T+2.71e11T2 |
| 47 | 1+(1.15e5−2.00e5i)T+(−2.53e11−4.38e11i)T2 |
| 53 | 1+(−9.93e5+5.73e5i)T+(5.87e11−1.01e12i)T2 |
| 59 | 1+(−2.38e5−4.13e5i)T+(−1.24e12+2.15e12i)T2 |
| 61 | 1+(9.92e5+5.72e5i)T+(1.57e12+2.72e12i)T2 |
| 67 | 1+(−1.88e6−3.26e6i)T+(−3.03e12+5.24e12i)T2 |
| 71 | 1+5.03e6iT−9.09e12T2 |
| 73 | 1+(−1.14e6+6.60e5i)T+(5.52e12−9.56e12i)T2 |
| 79 | 1+(1.53e6−2.65e6i)T+(−9.60e12−1.66e13i)T2 |
| 83 | 1−4.95e5T+2.71e13T2 |
| 89 | 1+(4.05e6−7.03e6i)T+(−2.21e13−3.83e13i)T2 |
| 97 | 1+1.70e7iT−8.07e13T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.74859425191714504198967526743, −11.29609771709750661768785280759, −10.37696900048091853006332908133, −9.283039449944762761114370174157, −8.558983561890886959532142933854, −6.88460465823449833761705979462, −5.22712583166778177672640540957, −3.22653566618497816251161603372, −1.31257032282796440128601838884, −0.26509394077804247633628127983,
1.93318910709638378530911475397, 3.96061207547006227159832935400, 6.40782712012112488895140267907, 6.79198910698315066558992052054, 8.568717859674796547599832081337, 9.367562793828724045819129595282, 10.32440115458052652220661290023, 11.90441773983100487957533466976, 13.04206132069980407598639978621, 14.56086628143917823863940553065