L(s) = 1 | + (1 + 1.73i)3-s + (48 − 83.1i)5-s + (119.5 − 206. i)9-s + (360 + 623. i)11-s + 572·13-s + 192·15-s + (−627 − 1.08e3i)17-s + (47 − 81.4i)19-s + (−48 + 83.1i)23-s + (−3.04e3 − 5.27e3i)25-s + 964·27-s − 4.37e3·29-s + (3.12e3 + 5.40e3i)31-s + (−720 + 1.24e3i)33-s + (5.39e3 − 9.35e3i)37-s + ⋯ |
L(s) = 1 | + (0.0641 + 0.111i)3-s + (0.858 − 1.48i)5-s + (0.491 − 0.851i)9-s + (0.897 + 1.55i)11-s + 0.938·13-s + 0.220·15-s + (−0.526 − 0.911i)17-s + (0.0298 − 0.0517i)19-s + (−0.0189 + 0.0327i)23-s + (−0.974 − 1.68i)25-s + 0.254·27-s − 0.965·29-s + (0.583 + 1.01i)31-s + (−0.115 + 0.199i)33-s + (0.648 − 1.12i)37-s + ⋯ |
Λ(s)=(=(196s/2ΓC(s)L(s)(0.386+0.922i)Λ(6−s)
Λ(s)=(=(196s/2ΓC(s+5/2)L(s)(0.386+0.922i)Λ(1−s)
Degree: |
2 |
Conductor: |
196
= 22⋅72
|
Sign: |
0.386+0.922i
|
Analytic conductor: |
31.4352 |
Root analytic conductor: |
5.60671 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ196(177,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 196, ( :5/2), 0.386+0.922i)
|
Particular Values
L(3) |
≈ |
2.642198392 |
L(21) |
≈ |
2.642198392 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
good | 3 | 1+(−1−1.73i)T+(−121.5+210.i)T2 |
| 5 | 1+(−48+83.1i)T+(−1.56e3−2.70e3i)T2 |
| 11 | 1+(−360−623.i)T+(−8.05e4+1.39e5i)T2 |
| 13 | 1−572T+3.71e5T2 |
| 17 | 1+(627+1.08e3i)T+(−7.09e5+1.22e6i)T2 |
| 19 | 1+(−47+81.4i)T+(−1.23e6−2.14e6i)T2 |
| 23 | 1+(48−83.1i)T+(−3.21e6−5.57e6i)T2 |
| 29 | 1+4.37e3T+2.05e7T2 |
| 31 | 1+(−3.12e3−5.40e3i)T+(−1.43e7+2.47e7i)T2 |
| 37 | 1+(−5.39e3+9.35e3i)T+(−3.46e7−6.00e7i)T2 |
| 41 | 1−1.20e4T+1.15e8T2 |
| 43 | 1+9.16e3T+1.47e8T2 |
| 47 | 1+(−1.29e4+2.23e4i)T+(−1.14e8−1.98e8i)T2 |
| 53 | 1+(507+878.i)T+(−2.09e8+3.62e8i)T2 |
| 59 | 1+(621+1.07e3i)T+(−3.57e8+6.19e8i)T2 |
| 61 | 1+(3.79e3−6.57e3i)T+(−4.22e8−7.31e8i)T2 |
| 67 | 1+(2.05e4+3.56e4i)T+(−6.75e8+1.16e9i)T2 |
| 71 | 1+3.76e4T+1.80e9T2 |
| 73 | 1+(−6.71e3−1.16e4i)T+(−1.03e9+1.79e9i)T2 |
| 79 | 1+(3.12e3−5.41e3i)T+(−1.53e9−2.66e9i)T2 |
| 83 | 1+2.52e4T+3.93e9T2 |
| 89 | 1+(−2.25e4+3.90e4i)T+(−2.79e9−4.83e9i)T2 |
| 97 | 1−1.07e5T+8.58e9T2 |
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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
−11.69892028731118300029891720455, −10.11822321942125499940221524956, −9.246517589269486315574353788281, −8.903823049369117879779130538055, −7.23787286664908900574329845401, −6.14327969513421458966312187946, −4.88413329179933057710171367319, −4.00948766188503505360097222881, −1.89389763388040805378306134700, −0.895393808723037845062778731143,
1.45585746379722354554727224778, 2.76775958912000936925935696216, 3.96232046359291411778283530961, 5.96517132056709824531322519630, 6.34816407409272334377585672761, 7.66335222581771956890716831223, 8.816814517174552640553453559451, 10.00834919721597987986457834928, 10.94867083935413610935020352043, 11.32464509502692729273788668847