L(s) = 1 | + (0.309 − 0.951i)3-s + (0.809 − 0.587i)5-s + (0.236 + 0.726i)7-s + (−0.809 − 0.587i)9-s + (3.04 + 1.31i)11-s + (4.73 + 3.44i)13-s + (−0.309 − 0.951i)15-s + (−6.16 + 4.47i)17-s + (−2.38 + 7.33i)19-s + 0.763·21-s + 7.61·23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.809 − 2.48i)29-s + (1.11 + 0.812i)31-s + ⋯ |
L(s) = 1 | + (0.178 − 0.549i)3-s + (0.361 − 0.262i)5-s + (0.0892 + 0.274i)7-s + (−0.269 − 0.195i)9-s + (0.918 + 0.396i)11-s + (1.31 + 0.954i)13-s + (−0.0797 − 0.245i)15-s + (−1.49 + 1.08i)17-s + (−0.546 + 1.68i)19-s + 0.166·21-s + 1.58·23-s + (0.0618 − 0.190i)25-s + (−0.155 + 0.113i)27-s + (−0.150 − 0.462i)29-s + (0.200 + 0.145i)31-s + ⋯ |
Λ(s)=(=(1320s/2ΓC(s)L(s)(0.970−0.242i)Λ(2−s)
Λ(s)=(=(1320s/2ΓC(s+1/2)L(s)(0.970−0.242i)Λ(1−s)
Degree: |
2 |
Conductor: |
1320
= 23⋅3⋅5⋅11
|
Sign: |
0.970−0.242i
|
Analytic conductor: |
10.5402 |
Root analytic conductor: |
3.24657 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1320(961,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1320, ( :1/2), 0.970−0.242i)
|
Particular Values
L(1) |
≈ |
1.994446314 |
L(21) |
≈ |
1.994446314 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(−0.309+0.951i)T |
| 5 | 1+(−0.809+0.587i)T |
| 11 | 1+(−3.04−1.31i)T |
good | 7 | 1+(−0.236−0.726i)T+(−5.66+4.11i)T2 |
| 13 | 1+(−4.73−3.44i)T+(4.01+12.3i)T2 |
| 17 | 1+(6.16−4.47i)T+(5.25−16.1i)T2 |
| 19 | 1+(2.38−7.33i)T+(−15.3−11.1i)T2 |
| 23 | 1−7.61T+23T2 |
| 29 | 1+(0.809+2.48i)T+(−23.4+17.0i)T2 |
| 31 | 1+(−1.11−0.812i)T+(9.57+29.4i)T2 |
| 37 | 1+(0.972+2.99i)T+(−29.9+21.7i)T2 |
| 41 | 1+(−0.854+2.62i)T+(−33.1−24.0i)T2 |
| 43 | 1+3.32T+43T2 |
| 47 | 1+(−0.572+1.76i)T+(−38.0−27.6i)T2 |
| 53 | 1+(−1.61−1.17i)T+(16.3+50.4i)T2 |
| 59 | 1+(−0.354−1.08i)T+(−47.7+34.6i)T2 |
| 61 | 1+(10.4−7.60i)T+(18.8−58.0i)T2 |
| 67 | 1−15.5T+67T2 |
| 71 | 1+(21.9−67.5i)T2 |
| 73 | 1+(2.14+6.60i)T+(−59.0+42.9i)T2 |
| 79 | 1+(−5.78−4.20i)T+(24.4+75.1i)T2 |
| 83 | 1+(−6.47+4.70i)T+(25.6−78.9i)T2 |
| 89 | 1−1.23T+89T2 |
| 97 | 1+(9.09+6.60i)T+(29.9+92.2i)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
−9.390271085412432312460156755001, −8.763251849656354018566742316092, −8.332378963019535327018655792956, −7.00974946816908279233160614095, −6.41124158539066140422942580259, −5.75467499838648262005227939669, −4.36898806183121812968621525665, −3.69389939103109713504884390591, −2.08216086008798193308182354172, −1.42925092985306223727873298812,
0.907268419458868295573186266966, 2.58461704059945877190505257873, 3.41671770822027582616929991666, 4.50239760415966084056603939491, 5.26652005991497531524159193121, 6.49858090769500093112824254072, 6.88203164629428727461525623226, 8.194119159991469157814075266723, 9.108485065154399847544996347908, 9.236956770503022320061335512593