L(s) = 1 | + (2 − 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (13 − 22.5i)5-s + 36·6-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (−51.9 − 90.0i)10-s + (179 + 310. i)11-s + (72 − 124. i)12-s − 332·13-s + 234·15-s + (−128 + 221. i)16-s + (63 + 109. i)17-s + (162 + 280. i)18-s + (−1.10e3 + 1.90e3i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.232 − 0.402i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.164 − 0.284i)10-s + (0.446 + 0.772i)11-s + (0.144 − 0.249i)12-s − 0.544·13-s + 0.268·15-s + (−0.125 + 0.216i)16-s + (0.0528 + 0.0915i)17-s + (0.117 + 0.204i)18-s + (−0.699 + 1.21i)19-s + ⋯ |
Λ(s)=(=(294s/2ΓC(s)L(s)(0.605−0.795i)Λ(6−s)
Λ(s)=(=(294s/2ΓC(s+5/2)L(s)(0.605−0.795i)Λ(1−s)
Degree: |
2 |
Conductor: |
294
= 2⋅3⋅72
|
Sign: |
0.605−0.795i
|
Analytic conductor: |
47.1528 |
Root analytic conductor: |
6.86679 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ294(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 294, ( :5/2), 0.605−0.795i)
|
Particular Values
L(3) |
≈ |
2.136566781 |
L(21) |
≈ |
2.136566781 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−2+3.46i)T |
| 3 | 1+(−4.5−7.79i)T |
| 7 | 1 |
good | 5 | 1+(−13+22.5i)T+(−1.56e3−2.70e3i)T2 |
| 11 | 1+(−179−310.i)T+(−8.05e4+1.39e5i)T2 |
| 13 | 1+332T+3.71e5T2 |
| 17 | 1+(−63−109.i)T+(−7.09e5+1.22e6i)T2 |
| 19 | 1+(1.10e3−1.90e3i)T+(−1.23e6−2.14e6i)T2 |
| 23 | 1+(−1.07e3+1.85e3i)T+(−3.21e6−5.57e6i)T2 |
| 29 | 1+3.61e3T+2.05e7T2 |
| 31 | 1+(−2.83e3−4.90e3i)T+(−1.43e7+2.47e7i)T2 |
| 37 | 1+(−1.46e3+2.53e3i)T+(−3.46e7−6.00e7i)T2 |
| 41 | 1−2.14e3T+1.15e8T2 |
| 43 | 1−6.38e3T+1.47e8T2 |
| 47 | 1+(3.26e3−5.64e3i)T+(−1.14e8−1.98e8i)T2 |
| 53 | 1+(−5.35e3−9.26e3i)T+(−2.09e8+3.62e8i)T2 |
| 59 | 1+(−2.12e4−3.68e4i)T+(−3.57e8+6.19e8i)T2 |
| 61 | 1+(2.24e4−3.88e4i)T+(−4.22e8−7.31e8i)T2 |
| 67 | 1+(−724−1.25e3i)T+(−6.75e8+1.16e9i)T2 |
| 71 | 1+4.40e3T+1.80e9T2 |
| 73 | 1+(−1.02e4−1.77e4i)T+(−1.03e9+1.79e9i)T2 |
| 79 | 1+(3.26e4−5.64e4i)T+(−1.53e9−2.66e9i)T2 |
| 83 | 1−1.02e5T+3.93e9T2 |
| 89 | 1+(6.40e4−1.10e5i)T+(−2.79e9−4.83e9i)T2 |
| 97 | 1−1.13e5T+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
−10.95972840037414925701694915414, −10.15188181225150597881321490842, −9.364826522892778695543790551364, −8.513954658983610688932399918090, −7.17846294992893375536625963588, −5.81949269244628833384782371554, −4.74644636969917032823404774823, −3.90387884159142652007025271696, −2.54445364563080335554005424394, −1.34688145532103827685278266608,
0.49602306900848312024214238818, 2.29781881329715037036176194058, 3.43198829009702849543240627499, 4.79561554778738951173206647394, 6.05793948831282596266349838456, 6.79804518967372025373211861673, 7.73713937743768409212497700691, 8.748365335055392703734936576431, 9.607192464078556703127653208561, 10.97247486440045530362241191083