L(s) = 1 | − 2-s + 3-s − 6-s + 7-s + 8-s + 3·9-s + 6·11-s + 8·13-s − 14-s − 16-s − 3·18-s − 2·19-s + 21-s − 6·22-s − 3·23-s + 24-s − 8·26-s + 8·27-s − 6·29-s − 8·31-s + 6·33-s − 4·37-s + 2·38-s + 8·39-s + 18·41-s − 42-s + 14·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 9-s + 1.80·11-s + 2.21·13-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s − 1.56·26-s + 1.53·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.657·37-s + 0.324·38-s + 1.28·39-s + 2.81·41-s − 0.154·42-s + 2.13·43-s + ⋯ |
Λ(s)=(=(122500s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(122500s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
122500
= 22⋅54⋅72
|
Sign: |
1
|
Analytic conductor: |
7.81070 |
Root analytic conductor: |
1.67175 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 122500, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.777786066 |
L(21) |
≈ |
1.777786066 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.44531500776178172422114943476, −11.14633118662792361884377904949, −10.80464514412187505352138054054, −10.40390860085647571050119551459, −9.646881200135278145608348813606, −9.189683285765849672773553132719, −9.003532390468188680012755658569, −8.708747181862142582449605468436, −7.918198123693549389279408758495, −7.73052133363762198184662013333, −6.99614545356602163056626935680, −6.56068623140851519642977026679, −5.99059187620313463522781212705, −5.53934080598421419226546320249, −4.36038292048665956218834927761, −3.95193641127438853822237163575, −3.87121871979297141974644355668, −2.71238625560086399033449734165, −1.46375846141625953745599028381, −1.37271047232170476058201585725,
1.37271047232170476058201585725, 1.46375846141625953745599028381, 2.71238625560086399033449734165, 3.87121871979297141974644355668, 3.95193641127438853822237163575, 4.36038292048665956218834927761, 5.53934080598421419226546320249, 5.99059187620313463522781212705, 6.56068623140851519642977026679, 6.99614545356602163056626935680, 7.73052133363762198184662013333, 7.918198123693549389279408758495, 8.708747181862142582449605468436, 9.003532390468188680012755658569, 9.189683285765849672773553132719, 9.646881200135278145608348813606, 10.40390860085647571050119551459, 10.80464514412187505352138054054, 11.14633118662792361884377904949, 11.44531500776178172422114943476