L(s) = 1 | + (−2.59 − 1.11i)2-s + (0.142 + 0.390i)3-s + (5.51 + 5.79i)4-s + (−20.2 − 3.57i)5-s + (0.0661 − 1.17i)6-s + (−16.1 + 9.30i)7-s + (−7.86 − 21.2i)8-s + (20.5 − 17.2i)9-s + (48.7 + 31.9i)10-s + (4.07 − 7.05i)11-s + (−1.48 + 2.98i)12-s + (26.9 + 9.81i)13-s + (52.2 − 6.20i)14-s + (−1.48 − 8.43i)15-s + (−3.23 + 63.9i)16-s + (32.2 + 27.0i)17-s + ⋯ |
L(s) = 1 | + (−0.918 − 0.394i)2-s + (0.0273 + 0.0752i)3-s + (0.689 + 0.724i)4-s + (−1.81 − 0.319i)5-s + (0.00450 − 0.0799i)6-s + (−0.870 + 0.502i)7-s + (−0.347 − 0.937i)8-s + (0.761 − 0.638i)9-s + (1.54 + 1.00i)10-s + (0.111 − 0.193i)11-s + (−0.0356 + 0.0716i)12-s + (0.575 + 0.209i)13-s + (0.997 − 0.118i)14-s + (−0.0256 − 0.145i)15-s + (−0.0504 + 0.998i)16-s + (0.460 + 0.386i)17-s + ⋯ |
Λ(s)=(=(152s/2ΓC(s)L(s)(0.998+0.0568i)Λ(4−s)
Λ(s)=(=(152s/2ΓC(s+3/2)L(s)(0.998+0.0568i)Λ(1−s)
Degree: |
2 |
Conductor: |
152
= 23⋅19
|
Sign: |
0.998+0.0568i
|
Analytic conductor: |
8.96829 |
Root analytic conductor: |
2.99471 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ152(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 152, ( :3/2), 0.998+0.0568i)
|
Particular Values
L(2) |
≈ |
0.678173−0.0192948i |
L(21) |
≈ |
0.678173−0.0192948i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(2.59+1.11i)T |
| 19 | 1+(57.1−59.9i)T |
good | 3 | 1+(−0.142−0.390i)T+(−20.6+17.3i)T2 |
| 5 | 1+(20.2+3.57i)T+(117.+42.7i)T2 |
| 7 | 1+(16.1−9.30i)T+(171.5−297.i)T2 |
| 11 | 1+(−4.07+7.05i)T+(−665.5−1.15e3i)T2 |
| 13 | 1+(−26.9−9.81i)T+(1.68e3+1.41e3i)T2 |
| 17 | 1+(−32.2−27.0i)T+(853.+4.83e3i)T2 |
| 23 | 1+(−179.+31.7i)T+(1.14e4−4.16e3i)T2 |
| 29 | 1+(−15.3+12.9i)T+(4.23e3−2.40e4i)T2 |
| 31 | 1+(−8.46−14.6i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1−235.T+5.06e4T2 |
| 41 | 1+(110.+303.i)T+(−5.27e4+4.43e4i)T2 |
| 43 | 1+(−12.2+69.3i)T+(−7.47e4−2.71e4i)T2 |
| 47 | 1+(−338.−403.i)T+(−1.80e4+1.02e5i)T2 |
| 53 | 1+(82.4+467.i)T+(−1.39e5+5.09e4i)T2 |
| 59 | 1+(45.7−54.5i)T+(−3.56e4−2.02e5i)T2 |
| 61 | 1+(709.−125.i)T+(2.13e5−7.76e4i)T2 |
| 67 | 1+(−430.−513.i)T+(−5.22e4+2.96e5i)T2 |
| 71 | 1+(127.−724.i)T+(−3.36e5−1.22e5i)T2 |
| 73 | 1+(−407.+148.i)T+(2.98e5−2.50e5i)T2 |
| 79 | 1+(−955.+347.i)T+(3.77e5−3.16e5i)T2 |
| 83 | 1+(122.+211.i)T+(−2.85e5+4.95e5i)T2 |
| 89 | 1+(327.−899.i)T+(−5.40e5−4.53e5i)T2 |
| 97 | 1+(−741.+884.i)T+(−1.58e5−8.98e5i)T2 |
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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
−12.47242120917577168078170943752, −11.48216026769049436067712767995, −10.54348832183299442774668961378, −9.245516757123247126939581987089, −8.523261885933464425184423989711, −7.45665174758246622047006495319, −6.43958173262325126729117684310, −4.10468480294144206484233567143, −3.25928993199813355347554086182, −0.842915964013817571501859472798,
0.69719634271811017733697512462, 3.15619552379525478994567715315, 4.63564643177794218575895333046, 6.66197984294798114187753080227, 7.32942152652500584967640448610, 8.133556119145631081531788294112, 9.356875701476139144005135480703, 10.62694958160685911841427725242, 11.16287137275724425904655330702, 12.36776790592187223279337529782