L(s) = 1 | + (2.63e6 − 3.26e6i)2-s + 5.11e9i·3-s + (−3.68e12 − 1.72e13i)4-s − 1.83e15·5-s + (1.66e16 + 1.34e16i)6-s − 1.24e18i·7-s + (−6.58e19 − 3.33e19i)8-s + 9.58e20·9-s + (−4.84e21 + 5.98e21i)10-s − 1.41e23i·11-s + (8.79e22 − 1.88e22i)12-s − 3.73e24·13-s + (−4.06e24 − 3.28e24i)14-s − 9.38e24i·15-s + (−2.82e26 + 1.26e26i)16-s + 9.62e26·17-s + ⋯ |
L(s) = 1 | + (0.628 − 0.777i)2-s + 0.162i·3-s + (−0.209 − 0.977i)4-s − 0.770·5-s + (0.126 + 0.102i)6-s − 0.318i·7-s + (−0.892 − 0.451i)8-s + 0.973·9-s + (−0.484 + 0.598i)10-s − 1.73i·11-s + (0.159 − 0.0341i)12-s − 1.16·13-s + (−0.247 − 0.200i)14-s − 0.125i·15-s + (−0.912 + 0.409i)16-s + 0.819·17-s + ⋯ |
Λ(s)=(=(4s/2ΓC(s)L(s)(−0.209−0.977i)Λ(45−s)
Λ(s)=(=(4s/2ΓC(s+22)L(s)(−0.209−0.977i)Λ(1−s)
Degree: |
2 |
Conductor: |
4
= 22
|
Sign: |
−0.209−0.977i
|
Analytic conductor: |
49.0478 |
Root analytic conductor: |
7.00342 |
Motivic weight: |
44 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ4(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 4, ( :22), −0.209−0.977i)
|
Particular Values
L(245) |
≈ |
0.3150298706 |
L(21) |
≈ |
0.3150298706 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−2.63e6+3.26e6i)T |
good | 3 | 1−5.11e9iT−9.84e20T2 |
| 5 | 1+1.83e15T+5.68e30T2 |
| 7 | 1+1.24e18iT−1.52e37T2 |
| 11 | 1+1.41e23iT−6.62e45T2 |
| 13 | 1+3.73e24T+1.03e49T2 |
| 17 | 1−9.62e26T+1.37e54T2 |
| 19 | 1−7.45e27iT−1.84e56T2 |
| 23 | 1−7.93e29iT−8.24e59T2 |
| 29 | 1+2.33e32T+2.21e64T2 |
| 31 | 1−4.48e31iT−4.16e65T2 |
| 37 | 1−1.89e34T+1.00e69T2 |
| 41 | 1+1.40e35T+9.17e70T2 |
| 43 | 1−4.14e35iT−7.45e71T2 |
| 47 | 1−6.22e36iT−3.73e73T2 |
| 53 | 1+5.66e37T+7.38e75T2 |
| 59 | 1−1.58e39iT−8.26e77T2 |
| 61 | 1+1.71e39T+3.58e78T2 |
| 67 | 1+1.52e39iT−2.22e80T2 |
| 71 | 1+5.26e40iT−2.85e81T2 |
| 73 | 1−1.04e40T+9.68e81T2 |
| 79 | 1+9.56e41iT−3.13e83T2 |
| 83 | 1−2.60e42iT−2.75e84T2 |
| 89 | 1+8.08e42T+5.93e85T2 |
| 97 | 1−1.49e43T+2.61e87T2 |
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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
−13.59963757794294174449701679085, −12.17918113185564577444764938711, −10.96634742737830741467507870658, −9.607627013765662916220607478734, −7.60762907406117819143479054566, −5.67268923073270960556232288887, −4.15433897659077121845474108859, −3.19767319663320608582352157741, −1.34338978916771176861350199351, −0.06861134077647163687549871883,
2.21412528086000797226096915695, 4.00915447488166447720199212903, 5.02051359277398581135902019997, 6.99118380974996308471205876936, 7.71071505849247338464671616416, 9.659059536732959385500987105281, 12.03395336168938900183308723869, 12.79686170870166627770982776401, 14.76435124379400235857499451726, 15.53386248257781804060050583490