L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−0.173 + 4.99i)5-s + (11.8 − 6.84i)7-s + 2.82·8-s + (6.24 − 3.32i)10-s + (10.6 − 6.15i)11-s + (14.7 + 8.50i)13-s + (−16.7 − 9.67i)14-s + (−2.00 − 3.46i)16-s + 6.89·17-s − 7.24·19-s + (−8.48 − 5.29i)20-s + (−15.0 − 8.70i)22-s + (−17.3 + 30.1i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0346 + 0.999i)5-s + (1.69 − 0.977i)7-s + 0.353·8-s + (0.624 − 0.332i)10-s + (0.969 − 0.559i)11-s + (1.13 + 0.654i)13-s + (−1.19 − 0.691i)14-s + (−0.125 − 0.216i)16-s + 0.405·17-s − 0.381·19-s + (−0.424 − 0.264i)20-s + (−0.685 − 0.395i)22-s + (−0.756 + 1.31i)23-s + ⋯ |
Λ(s)=(=(810s/2ΓC(s)L(s)(0.927+0.374i)Λ(3−s)
Λ(s)=(=(810s/2ΓC(s+1)L(s)(0.927+0.374i)Λ(1−s)
Degree: |
2 |
Conductor: |
810
= 2⋅34⋅5
|
Sign: |
0.927+0.374i
|
Analytic conductor: |
22.0709 |
Root analytic conductor: |
4.69796 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ810(539,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 810, ( :1), 0.927+0.374i)
|
Particular Values
L(23) |
≈ |
2.054798167 |
L(21) |
≈ |
2.054798167 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.707+1.22i)T |
| 3 | 1 |
| 5 | 1+(0.173−4.99i)T |
good | 7 | 1+(−11.8+6.84i)T+(24.5−42.4i)T2 |
| 11 | 1+(−10.6+6.15i)T+(60.5−104.i)T2 |
| 13 | 1+(−14.7−8.50i)T+(84.5+146.i)T2 |
| 17 | 1−6.89T+289T2 |
| 19 | 1+7.24T+361T2 |
| 23 | 1+(17.3−30.1i)T+(−264.5−458.i)T2 |
| 29 | 1+(18.3−10.5i)T+(420.5−728.i)T2 |
| 31 | 1+(−19.1+33.0i)T+(−480.5−832.i)T2 |
| 37 | 1+21.5iT−1.36e3T2 |
| 41 | 1+(31.4+18.1i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(−5.40+3.11i)T+(924.5−1.60e3i)T2 |
| 47 | 1+(−20.1−34.8i)T+(−1.10e3+1.91e3i)T2 |
| 53 | 1−38.2T+2.80e3T2 |
| 59 | 1+(−36.0−20.8i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(−7.52−13.0i)T+(−1.86e3+3.22e3i)T2 |
| 67 | 1+(−111.−64.3i)T+(2.24e3+3.88e3i)T2 |
| 71 | 1+104.iT−5.04e3T2 |
| 73 | 1−2.11iT−5.32e3T2 |
| 79 | 1+(−22.0−38.1i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(−27.5−47.6i)T+(−3.44e3+5.96e3i)T2 |
| 89 | 1−68.1iT−7.92e3T2 |
| 97 | 1+(−87.6+50.6i)T+(4.70e3−8.14e3i)T2 |
show more | |
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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
−10.23150001782819021307245110250, −9.212727070228278294473328050658, −8.250480398121873329902629198779, −7.61123076819140627103038629897, −6.70567751045605391798151332346, −5.57968622060594828872829359910, −4.00927154651713340206162425555, −3.77756894658837106240983324333, −2.03140740518457969547061455779, −1.12439497969789112115104229166,
1.07220884815496910694421035658, 1.99201252381411382649445591788, 4.04741587646894709786066585647, 4.91103450413610078770956029919, 5.62336141031455296229460593563, 6.55835636843376005729585782665, 7.934363852774435211462295239629, 8.431838865446979677784683448950, 8.839621132032531590312986708550, 9.912592827963130609713457412118