L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.500 − 0.866i)4-s + (−2.59 − 1.5i)7-s + 3i·8-s + (−1 + 1.73i)11-s + (−1.73 + i)13-s + (1.5 + 2.59i)14-s + (0.500 − 0.866i)16-s + 4i·17-s + 8·19-s + (1.73 − 0.999i)22-s + (−2.59 + 1.5i)23-s + 1.99·26-s + 3i·28-s + (0.5 − 0.866i)29-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.250 − 0.433i)4-s + (−0.981 − 0.566i)7-s + 1.06i·8-s + (−0.301 + 0.522i)11-s + (−0.480 + 0.277i)13-s + (0.400 + 0.694i)14-s + (0.125 − 0.216i)16-s + 0.970i·17-s + 1.83·19-s + (0.369 − 0.213i)22-s + (−0.541 + 0.312i)23-s + 0.392·26-s + 0.566i·28-s + (0.0928 − 0.160i)29-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(0.687−0.726i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(0.687−0.726i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
0.687−0.726i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(424,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), 0.687−0.726i)
|
Particular Values
L(1) |
≈ |
0.529410+0.227808i |
L(21) |
≈ |
0.529410+0.227808i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
good | 2 | 1+(0.866+0.5i)T+(1+1.73i)T2 |
| 7 | 1+(2.59+1.5i)T+(3.5+6.06i)T2 |
| 11 | 1+(1−1.73i)T+(−5.5−9.52i)T2 |
| 13 | 1+(1.73−i)T+(6.5−11.2i)T2 |
| 17 | 1−4iT−17T2 |
| 19 | 1−8T+19T2 |
| 23 | 1+(2.59−1.5i)T+(11.5−19.9i)T2 |
| 29 | 1+(−0.5+0.866i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−15.5+26.8i)T2 |
| 37 | 1−4iT−37T2 |
| 41 | 1+(−2.5−4.33i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−6.92−4i)T+(21.5+37.2i)T2 |
| 47 | 1+(6.06+3.5i)T+(23.5+40.7i)T2 |
| 53 | 1−2iT−53T2 |
| 59 | 1+(−7−12.1i)T+(−29.5+51.0i)T2 |
| 61 | 1+(3.5−6.06i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−2.59+1.5i)T+(33.5−58.0i)T2 |
| 71 | 1+2T+71T2 |
| 73 | 1−4iT−73T2 |
| 79 | 1+(3−5.19i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−7.79−4.5i)T+(41.5+71.8i)T2 |
| 89 | 1+15T+89T2 |
| 97 | 1+(−1.73−i)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
−10.22407999000492943217883549543, −9.887697750403088476177685671307, −9.215358756927379140123313675578, −8.070132340964677675155044062206, −7.24175662027074163879985833183, −6.14821909496671207394796014582, −5.19054047779975664598592125918, −4.03289701584901501325640499204, −2.71976817588939256949472524394, −1.24308645829007819880232275434,
0.42353469952578296085656667924, 2.77109530669204589411747202351, 3.56402857927525764084556600102, 5.03556487538940634643289030534, 6.02317889951852103025632616963, 7.11118042648535941362385945832, 7.73778310422818767144483377660, 8.738211807504768224312620157485, 9.522507313171193712129945389851, 9.920074968257708355999846022191