L(s) = 1 | + (4 + 4i)2-s + 32i·4-s + (362. + 362. i)7-s + (−128 + 128i)8-s + 1.32e3·11-s + (−254. + 254. i)13-s + 2.90e3i·14-s − 1.02e3·16-s + (4.39e3 + 4.39e3i)17-s − 819. i·19-s + (5.28e3 + 5.28e3i)22-s + (1.11e4 − 1.11e4i)23-s − 2.03e3·26-s + (−1.16e4 + 1.16e4i)28-s + 2.74e4i·29-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + 0.5i·4-s + (1.05 + 1.05i)7-s + (−0.250 + 0.250i)8-s + 0.992·11-s + (−0.115 + 0.115i)13-s + 1.05i·14-s − 0.250·16-s + (0.895 + 0.895i)17-s − 0.119i·19-s + (0.496 + 0.496i)22-s + (0.915 − 0.915i)23-s − 0.115·26-s + (−0.528 + 0.528i)28-s + 1.12i·29-s + ⋯ |
Λ(s)=(=(450s/2ΓC(s)L(s)(−0.229−0.973i)Λ(7−s)
Λ(s)=(=(450s/2ΓC(s+3)L(s)(−0.229−0.973i)Λ(1−s)
Degree: |
2 |
Conductor: |
450
= 2⋅32⋅52
|
Sign: |
−0.229−0.973i
|
Analytic conductor: |
103.524 |
Root analytic conductor: |
10.1746 |
Motivic weight: |
6 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ450(307,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 450, ( :3), −0.229−0.973i)
|
Particular Values
L(27) |
≈ |
3.901708181 |
L(21) |
≈ |
3.901708181 |
L(4) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−4−4i)T |
| 3 | 1 |
| 5 | 1 |
good | 7 | 1+(−362.−362.i)T+1.17e5iT2 |
| 11 | 1−1.32e3T+1.77e6T2 |
| 13 | 1+(254.−254.i)T−4.82e6iT2 |
| 17 | 1+(−4.39e3−4.39e3i)T+2.41e7iT2 |
| 19 | 1+819.iT−4.70e7T2 |
| 23 | 1+(−1.11e4+1.11e4i)T−1.48e8iT2 |
| 29 | 1−2.74e4iT−5.94e8T2 |
| 31 | 1−1.50e4T+8.87e8T2 |
| 37 | 1+(2.15e4+2.15e4i)T+2.56e9iT2 |
| 41 | 1−4.56e4T+4.75e9T2 |
| 43 | 1+(−7.81e4+7.81e4i)T−6.32e9iT2 |
| 47 | 1+(1.99e3+1.99e3i)T+1.07e10iT2 |
| 53 | 1+(−1.06e5+1.06e5i)T−2.21e10iT2 |
| 59 | 1+6.44e4iT−4.21e10T2 |
| 61 | 1+3.20e5T+5.15e10T2 |
| 67 | 1+(−3.50e4−3.50e4i)T+9.04e10iT2 |
| 71 | 1−2.51e5T+1.28e11T2 |
| 73 | 1+(3.26e5−3.26e5i)T−1.51e11iT2 |
| 79 | 1+2.57e4iT−2.43e11T2 |
| 83 | 1+(1.23e5−1.23e5i)T−3.26e11iT2 |
| 89 | 1+1.98e5iT−4.96e11T2 |
| 97 | 1+(−7.38e5−7.38e5i)T+8.32e11iT2 |
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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
−10.49899875136955644813774903029, −9.063149947254500167899701137658, −8.599920332186315433056078815896, −7.58394142126644643908830394758, −6.53669355065361700236131997863, −5.60041231498892687565561022108, −4.80049265550793102491263147102, −3.70252776374047082903246899760, −2.39743120110000453659291709781, −1.21797566751184326770236933387,
0.804183475403028501775609627457, 1.47940989412192124501967383082, 2.96700992366751523280767277000, 4.08475045999109777530163188242, 4.80804246813951560590907938710, 5.91570995710097906281908362720, 7.16195738301155980816298033170, 7.84948522454693229051838798935, 9.164415648449143602480766607887, 9.974790004982237691257584688752