L(s) = 1 | + (−1.29 + 2.24i)2-s + (−2.34 − 4.06i)4-s + 2.28·5-s + (−2.08 − 1.63i)7-s + 6.97·8-s + (−2.95 + 5.11i)10-s − 2.95·11-s + (−2.13 + 3.69i)13-s + (6.34 − 2.56i)14-s + (−4.32 + 7.49i)16-s + (0.764 − 1.32i)17-s + (−3.69 − 6.39i)19-s + (−5.35 − 9.28i)20-s + (3.82 − 6.62i)22-s − 6.15·23-s + ⋯ |
L(s) = 1 | + (−0.914 + 1.58i)2-s + (−1.17 − 2.03i)4-s + 1.02·5-s + (−0.787 − 0.616i)7-s + 2.46·8-s + (−0.934 + 1.61i)10-s − 0.891·11-s + (−0.591 + 1.02i)13-s + (1.69 − 0.684i)14-s + (−1.08 + 1.87i)16-s + (0.185 − 0.321i)17-s + (−0.846 − 1.46i)19-s + (−1.19 − 2.07i)20-s + (0.815 − 1.41i)22-s − 1.28·23-s + ⋯ |
Λ(s)=(=(567s/2ΓC(s)L(s)(0.474+0.880i)Λ(2−s)
Λ(s)=(=(567s/2ΓC(s+1/2)L(s)(0.474+0.880i)Λ(1−s)
Degree: |
2 |
Conductor: |
567
= 34⋅7
|
Sign: |
0.474+0.880i
|
Analytic conductor: |
4.52751 |
Root analytic conductor: |
2.12779 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ567(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 567, ( :1/2), 0.474+0.880i)
|
Particular Values
L(1) |
≈ |
0.206908−0.123509i |
L(21) |
≈ |
0.206908−0.123509i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(2.08+1.63i)T |
good | 2 | 1+(1.29−2.24i)T+(−1−1.73i)T2 |
| 5 | 1−2.28T+5T2 |
| 11 | 1+2.95T+11T2 |
| 13 | 1+(2.13−3.69i)T+(−6.5−11.2i)T2 |
| 17 | 1+(−0.764+1.32i)T+(−8.5−14.7i)T2 |
| 19 | 1+(3.69+6.39i)T+(−9.5+16.4i)T2 |
| 23 | 1+6.15T+23T2 |
| 29 | 1+(−1.17−2.02i)T+(−14.5+25.1i)T2 |
| 31 | 1+(3.11+5.38i)T+(−15.5+26.8i)T2 |
| 37 | 1+(3.58+6.21i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−3.94+6.83i)T+(−20.5−35.5i)T2 |
| 43 | 1+(0.417+0.722i)T+(−21.5+37.2i)T2 |
| 47 | 1+(2.91−5.04i)T+(−23.5−40.7i)T2 |
| 53 | 1+(3.71−6.44i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−2.31−4.00i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−3.56+6.17i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−1.66−2.87i)T+(−33.5+58.0i)T2 |
| 71 | 1−0.160T+71T2 |
| 73 | 1+(−0.190+0.329i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−3.97+6.88i)T+(−39.5−68.4i)T2 |
| 83 | 1+(2.14+3.72i)T+(−41.5+71.8i)T2 |
| 89 | 1+(−3.02−5.24i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−0.661−1.14i)T+(−48.5+84.0i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.13157261686031503306013795956, −9.486778985976792623575299908520, −8.949342026809977178739443277229, −7.70864135860045406087514511902, −7.02382245588953677847017341399, −6.26791318496112858642434691737, −5.47141299191055090886011923026, −4.37605729086767905638561359002, −2.22175572953440850901330972208, −0.17216227046461902905540869565,
1.81987741576025857086640531223, 2.66073305498472674897598189038, 3.65695191168223132791147927480, 5.28393469457247548529574255192, 6.26289742640017041865197920180, 7.930123131962110496063968360083, 8.448051118797600015026515261813, 9.622748764108280806936781684594, 10.14012119807402988016608843101, 10.40705756575758765177965035743