L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.133 + 2.23i)5-s + (−1.73 + i)7-s + 0.999i·8-s + (−1 − 1.99i)10-s + (1 + 1.73i)11-s + (−5.19 − 3i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s + 2i·17-s + (1.86 + 1.23i)20-s + (−1.73 − 0.999i)22-s + (−3.46 − 2i)23-s + (−4.96 − 0.598i)25-s + 6·26-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0599 + 0.998i)5-s + (−0.654 + 0.377i)7-s + 0.353i·8-s + (−0.316 − 0.632i)10-s + (0.301 + 0.522i)11-s + (−1.44 − 0.832i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + 0.485i·17-s + (0.417 + 0.275i)20-s + (−0.369 − 0.213i)22-s + (−0.722 − 0.417i)23-s + (−0.992 − 0.119i)25-s + 1.17·26-s + ⋯ |
Λ(s)=(=(810s/2ΓC(s)L(s)(−0.803+0.595i)Λ(2−s)
Λ(s)=(=(810s/2ΓC(s+1/2)L(s)(−0.803+0.595i)Λ(1−s)
Degree: |
2 |
Conductor: |
810
= 2⋅34⋅5
|
Sign: |
−0.803+0.595i
|
Analytic conductor: |
6.46788 |
Root analytic conductor: |
2.54320 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ810(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 810, ( :1/2), −0.803+0.595i)
|
Particular Values
L(1) |
≈ |
0.0495725−0.150047i |
L(21) |
≈ |
0.0495725−0.150047i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.866−0.5i)T |
| 3 | 1 |
| 5 | 1+(0.133−2.23i)T |
good | 7 | 1+(1.73−i)T+(3.5−6.06i)T2 |
| 11 | 1+(−1−1.73i)T+(−5.5+9.52i)T2 |
| 13 | 1+(5.19+3i)T+(6.5+11.2i)T2 |
| 17 | 1−2iT−17T2 |
| 19 | 1+19T2 |
| 23 | 1+(3.46+2i)T+(11.5+19.9i)T2 |
| 29 | 1+(−14.5+25.1i)T2 |
| 31 | 1+(−4+6.92i)T+(−15.5−26.8i)T2 |
| 37 | 1+2iT−37T2 |
| 41 | 1+(−1+1.73i)T+(−20.5−35.5i)T2 |
| 43 | 1+(3.46−2i)T+(21.5−37.2i)T2 |
| 47 | 1+(6.92−4i)T+(23.5−40.7i)T2 |
| 53 | 1+6iT−53T2 |
| 59 | 1+(5−8.66i)T+(−29.5−51.0i)T2 |
| 61 | 1+(1+1.73i)T+(−30.5+52.8i)T2 |
| 67 | 1+(6.92+4i)T+(33.5+58.0i)T2 |
| 71 | 1+12T+71T2 |
| 73 | 1+4iT−73T2 |
| 79 | 1+(−39.5+68.4i)T2 |
| 83 | 1+(−3.46+2i)T+(41.5−71.8i)T2 |
| 89 | 1+10T+89T2 |
| 97 | 1+(−6.92+4i)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
−10.35443652003373093948879808569, −9.989861149836485816101968818152, −9.237050423315198991024725686187, −7.990012882917082860832994589526, −7.43395902468856121834926917469, −6.47949295941319424134878707995, −5.85438300087007503244714141412, −4.52330383227847479585652771621, −3.10341379116708057205929908035, −2.17890744469028945495890594484,
0.091413956349506470307452914209, 1.59464547805471948132622686554, 3.01972959333313213593525001656, 4.20432973635725800949346151524, 5.12939463696529160850278545433, 6.43385647378588617098655425467, 7.26943802593790705951199392955, 8.204656042193831365106203067425, 9.053470736315173201399896386020, 9.691650129231506761982155647375