L(s) = 1 | + (−0.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 + 1.73i)11-s + 1.99·12-s + (−3.5 − 0.866i)13-s + (−1.99 − 3.46i)16-s + (3 + 1.73i)19-s − 1.73i·21-s + (−3 − 5.19i)23-s + 0.999·27-s + (−3 + 1.73i)28-s + (−3 − 5.19i)29-s − 1.73i·31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.566 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 + 0.522i)11-s + 0.577·12-s + (−0.970 − 0.240i)13-s + (−0.499 − 0.866i)16-s + (0.688 + 0.397i)19-s − 0.377i·21-s + (−0.625 − 1.08i)23-s + 0.192·27-s + (−0.566 + 0.327i)28-s + (−0.557 − 0.964i)29-s − 0.311i·31-s + ⋯ |
Λ(s)=(=(975s/2ΓC(s)L(s)(−0.964+0.265i)Λ(2−s)
Λ(s)=(=(975s/2ΓC(s+1/2)L(s)(−0.964+0.265i)Λ(1−s)
Degree: |
2 |
Conductor: |
975
= 3⋅52⋅13
|
Sign: |
−0.964+0.265i
|
Analytic conductor: |
7.78541 |
Root analytic conductor: |
2.79023 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ975(751,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 975, ( :1/2), −0.964+0.265i)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.5+0.866i)T |
| 5 | 1 |
| 13 | 1+(3.5+0.866i)T |
good | 2 | 1+(1−1.73i)T2 |
| 7 | 1+(−1.5−0.866i)T+(3.5+6.06i)T2 |
| 11 | 1+(3−1.73i)T+(5.5−9.52i)T2 |
| 17 | 1+(−8.5−14.7i)T2 |
| 19 | 1+(−3−1.73i)T+(9.5+16.4i)T2 |
| 23 | 1+(3+5.19i)T+(−11.5+19.9i)T2 |
| 29 | 1+(3+5.19i)T+(−14.5+25.1i)T2 |
| 31 | 1+1.73iT−31T2 |
| 37 | 1+(18.5−32.0i)T2 |
| 41 | 1+(6−3.46i)T+(20.5−35.5i)T2 |
| 43 | 1+(0.5−0.866i)T+(−21.5−37.2i)T2 |
| 47 | 1−3.46iT−47T2 |
| 53 | 1+12T+53T2 |
| 59 | 1+(3+1.73i)T+(29.5+51.0i)T2 |
| 61 | 1+(0.5−0.866i)T+(−30.5−52.8i)T2 |
| 67 | 1+(7.5−4.33i)T+(33.5−58.0i)T2 |
| 71 | 1+(9+5.19i)T+(35.5+61.4i)T2 |
| 73 | 1−1.73iT−73T2 |
| 79 | 1+11T+79T2 |
| 83 | 1+13.8iT−83T2 |
| 89 | 1+(−6+3.46i)T+(44.5−77.0i)T2 |
| 97 | 1+(−4.5−2.59i)T+(48.5+84.0i)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
−9.651715615506368687536095045272, −8.584250861254240999058392445136, −7.75599880232631505058479938482, −7.50064747390494244364191570336, −6.18808933732839416372932720253, −5.08615404826134296847720932980, −4.49952413052877564035388332568, −3.04759202969689392374417843759, −2.06555189694570825403784157666, 0,
1.62653516659400084362747731031, 3.21632049637395201077392991657, 4.48956763572689170067062061150, 5.17307383378504087078451922997, 5.75750726649752517900508221771, 7.02286014279295953272144916573, 7.911030600551024386461923548373, 8.939055122074977073629508138444, 9.670613633660325487730464257980