L(s) = 1 | + (1.93 − 0.517i)2-s + (0.599 − 1.62i)3-s + (1.73 − 0.999i)4-s + (1.30 − 1.81i)5-s + (0.317 − 3.44i)6-s + (−2.28 − 1.94i)9-s + (1.58 − 4.18i)10-s + (−0.949 + 0.548i)11-s + (−0.585 − 3.41i)12-s + (−2.43 − 2.43i)13-s + (−2.16 − 3.21i)15-s + (−1.99 + 3.46i)16-s + (0.978 − 3.65i)17-s + (−5.41 − 2.58i)18-s + (6.67 + 3.85i)19-s + (0.449 − 4.44i)20-s + ⋯ |
L(s) = 1 | + (1.36 − 0.366i)2-s + (0.346 − 0.938i)3-s + (0.866 − 0.499i)4-s + (0.584 − 0.811i)5-s + (0.129 − 1.40i)6-s + (−0.760 − 0.649i)9-s + (0.501 − 1.32i)10-s + (−0.286 + 0.165i)11-s + (−0.169 − 0.985i)12-s + (−0.676 − 0.676i)13-s + (−0.558 − 0.829i)15-s + (−0.499 + 0.866i)16-s + (0.237 − 0.886i)17-s + (−1.27 − 0.609i)18-s + (1.53 + 0.884i)19-s + (0.100 − 0.994i)20-s + ⋯ |
Λ(s)=(=(735s/2ΓC(s)L(s)(−0.281+0.959i)Λ(2−s)
Λ(s)=(=(735s/2ΓC(s+1/2)L(s)(−0.281+0.959i)Λ(1−s)
Degree: |
2 |
Conductor: |
735
= 3⋅5⋅72
|
Sign: |
−0.281+0.959i
|
Analytic conductor: |
5.86900 |
Root analytic conductor: |
2.42260 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ735(128,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 735, ( :1/2), −0.281+0.959i)
|
Particular Values
L(1) |
≈ |
2.09086−2.79119i |
L(21) |
≈ |
2.09086−2.79119i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.599+1.62i)T |
| 5 | 1+(−1.30+1.81i)T |
| 7 | 1 |
good | 2 | 1+(−1.93+0.517i)T+(1.73−i)T2 |
| 11 | 1+(0.949−0.548i)T+(5.5−9.52i)T2 |
| 13 | 1+(2.43+2.43i)T+13iT2 |
| 17 | 1+(−0.978+3.65i)T+(−14.7−8.5i)T2 |
| 19 | 1+(−6.67−3.85i)T+(9.5+16.4i)T2 |
| 23 | 1+(−1.15−4.29i)T+(−19.9+11.5i)T2 |
| 29 | 1−8.66T+29T2 |
| 31 | 1+(−1.73−3i)T+(−15.5+26.8i)T2 |
| 37 | 1+(−1.26−4.71i)T+(−32.0+18.5i)T2 |
| 41 | 1+6.89iT−41T2 |
| 43 | 1+(−1.89−1.89i)T+43iT2 |
| 47 | 1+(3.65−0.978i)T+(40.7−23.5i)T2 |
| 53 | 1+(2.80+0.750i)T+(45.8+26.5i)T2 |
| 59 | 1+(5+8.66i)T+(−29.5+51.0i)T2 |
| 61 | 1+(4.24−7.34i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−2.11−0.567i)T+(58.0+33.5i)T2 |
| 71 | 1+9.75iT−71T2 |
| 73 | 1+(2.58−9.65i)T+(−63.2−36.5i)T2 |
| 79 | 1+(−10.3−5.94i)T+(39.5+68.4i)T2 |
| 83 | 1+(5.34−5.34i)T−83iT2 |
| 89 | 1+(−8.44+14.6i)T+(−44.5−77.0i)T2 |
| 97 | 1+(2.43−2.43i)T−97iT2 |
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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
−10.12481032021981169511333749444, −9.319453707964826123750144739446, −8.265169150506873243327142657270, −7.45775556683436935279338495239, −6.30186358036590259498092513871, −5.38697225866169219643076925229, −4.89766350382330375562913854731, −3.34759962138589454806974041856, −2.58121660507848731643836444942, −1.24936558235640218963221052177,
2.57096309320958123949871810393, 3.20452751152991810447014531309, 4.37472055351778689421940855552, 5.08355068172706067744609928818, 6.00194043241585709972128930401, 6.80593335945391098317002032888, 7.85120730130154041815782686192, 9.166485624839321319965628651800, 9.803742303467096887971839076556, 10.65302056052512280430025979651