L(s) = 1 | + (0.701 + 1.21i)2-s + (−1.5 − 0.866i)3-s + (0.0157 − 0.0272i)4-s + (1.93 − 1.11i)5-s − 2.43i·6-s + 2.85·8-s + (1.5 + 2.59i)9-s + (2.71 + 1.56i)10-s + (−0.0472 + 0.0272i)12-s − 3.87·15-s + (1.96 + 3.40i)16-s + (−6.98 − 4.03i)17-s + (−2.10 + 3.64i)18-s + (5.76 − 3.32i)19-s − 0.0704i·20-s + ⋯ |
L(s) = 1 | + (0.496 + 0.859i)2-s + (−0.866 − 0.499i)3-s + (0.00787 − 0.0136i)4-s + (0.866 − 0.499i)5-s − 0.992i·6-s + 1.00·8-s + (0.5 + 0.866i)9-s + (0.859 + 0.496i)10-s + (−0.0136 + 0.00787i)12-s − 1.00·15-s + (0.491 + 0.852i)16-s + (−1.69 − 0.977i)17-s + (−0.496 + 0.859i)18-s + (1.32 − 0.763i)19-s − 0.0157i·20-s + ⋯ |
Λ(s)=(=(735s/2ΓC(s)L(s)(0.982+0.188i)Λ(2−s)
Λ(s)=(=(735s/2ΓC(s+1/2)L(s)(0.982+0.188i)Λ(1−s)
Degree: |
2 |
Conductor: |
735
= 3⋅5⋅72
|
Sign: |
0.982+0.188i
|
Analytic conductor: |
5.86900 |
Root analytic conductor: |
2.42260 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ735(374,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 735, ( :1/2), 0.982+0.188i)
|
Particular Values
L(1) |
≈ |
1.91895−0.182513i |
L(21) |
≈ |
1.91895−0.182513i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.5+0.866i)T |
| 5 | 1+(−1.93+1.11i)T |
| 7 | 1 |
good | 2 | 1+(−0.701−1.21i)T+(−1+1.73i)T2 |
| 11 | 1+(5.5+9.52i)T2 |
| 13 | 1+13T2 |
| 17 | 1+(6.98+4.03i)T+(8.5+14.7i)T2 |
| 19 | 1+(−5.76+3.32i)T+(9.5−16.4i)T2 |
| 23 | 1+(0.111+0.192i)T+(−11.5+19.9i)T2 |
| 29 | 1−29T2 |
| 31 | 1+(−8.84−5.10i)T+(15.5+26.8i)T2 |
| 37 | 1+(18.5−32.0i)T2 |
| 41 | 1+41T2 |
| 43 | 1−43T2 |
| 47 | 1+(0.886−0.511i)T+(23.5−40.7i)T2 |
| 53 | 1+(−5.54+9.60i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−29.5−51.0i)T2 |
| 61 | 1+(13.0−7.53i)T+(30.5−52.8i)T2 |
| 67 | 1+(33.5+58.0i)T2 |
| 71 | 1−71T2 |
| 73 | 1+(−36.5−63.2i)T2 |
| 79 | 1+(2.91+5.05i)T+(−39.5+68.4i)T2 |
| 83 | 1−15.0iT−83T2 |
| 89 | 1+(−44.5+77.0i)T2 |
| 97 | 1+97T2 |
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
−10.40619543048951888408385927793, −9.550011786532002029413263756167, −8.487402118520978122412752119478, −7.27424949617216490460142953521, −6.70864321116137886435947618551, −5.97024156465507578616722291308, −5.00707427160934197894986710117, −4.67238919055432237512579345245, −2.41012262124259710933505301600, −1.06944182156322602394991522636,
1.55599893875830309781945744555, 2.80088856083354210935939408992, 3.92599101477187146245069125615, 4.78602030502439427327725364607, 5.88789743199492640759808375927, 6.60183819434215832765471560444, 7.68037992021194934683674501212, 9.059555977474144378979279061003, 10.01715317723032320004227329080, 10.50703337102210298161302633942