L(s) = 1 | − 1.69·2-s + (0.175 + 1.72i)3-s + 0.888·4-s + (0.474 − 0.822i)5-s + (−0.298 − 2.92i)6-s + 1.88·8-s + (−2.93 + 0.605i)9-s + (−0.806 + 1.39i)10-s + (0.294 + 0.509i)11-s + (0.156 + 1.53i)12-s + (2.50 + 4.34i)13-s + (1.5 + 0.673i)15-s − 4.98·16-s + (3.79 − 6.56i)17-s + (4.99 − 1.02i)18-s + (2.23 + 3.86i)19-s + ⋯ |
L(s) = 1 | − 1.20·2-s + (0.101 + 0.994i)3-s + 0.444·4-s + (0.212 − 0.367i)5-s + (−0.121 − 1.19i)6-s + 0.667·8-s + (−0.979 + 0.201i)9-s + (−0.255 + 0.441i)10-s + (0.0886 + 0.153i)11-s + (0.0451 + 0.442i)12-s + (0.696 + 1.20i)13-s + (0.387 + 0.173i)15-s − 1.24·16-s + (0.919 − 1.59i)17-s + (1.17 − 0.242i)18-s + (0.511 + 0.886i)19-s + ⋯ |
Λ(s)=(=(441s/2ΓC(s)L(s)(−0.154−0.987i)Λ(2−s)
Λ(s)=(=(441s/2ΓC(s+1/2)L(s)(−0.154−0.987i)Λ(1−s)
Degree: |
2 |
Conductor: |
441
= 32⋅72
|
Sign: |
−0.154−0.987i
|
Analytic conductor: |
3.52140 |
Root analytic conductor: |
1.87654 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ441(214,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 441, ( :1/2), −0.154−0.987i)
|
Particular Values
L(1) |
≈ |
0.458210+0.535682i |
L(21) |
≈ |
0.458210+0.535682i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.175−1.72i)T |
| 7 | 1 |
good | 2 | 1+1.69T+2T2 |
| 5 | 1+(−0.474+0.822i)T+(−2.5−4.33i)T2 |
| 11 | 1+(−0.294−0.509i)T+(−5.5+9.52i)T2 |
| 13 | 1+(−2.50−4.34i)T+(−6.5+11.2i)T2 |
| 17 | 1+(−3.79+6.56i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−2.23−3.86i)T+(−9.5+16.4i)T2 |
| 23 | 1+(1.23−2.14i)T+(−11.5−19.9i)T2 |
| 29 | 1+(2.73−4.74i)T+(−14.5−25.1i)T2 |
| 31 | 1+6.07T+31T2 |
| 37 | 1+(−3.49−6.05i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−0.527−0.913i)T+(−20.5+35.5i)T2 |
| 43 | 1+(3.49−6.05i)T+(−21.5−37.2i)T2 |
| 47 | 1+7.47T+47T2 |
| 53 | 1+(3.46−5.99i)T+(−26.5−45.8i)T2 |
| 59 | 1−10.4T+59T2 |
| 61 | 1−11.6T+61T2 |
| 67 | 1+11.8T+67T2 |
| 71 | 1−4.30T+71T2 |
| 73 | 1+(2.23−3.86i)T+(−36.5−63.2i)T2 |
| 79 | 1+1.33T+79T2 |
| 83 | 1+(−2.84+4.92i)T+(−41.5−71.8i)T2 |
| 89 | 1+(0.421+0.730i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−1.70+2.94i)T+(−48.5−84.0i)T2 |
show more | |
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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
−11.22116197663704349037386080188, −10.05128031632673280274990252485, −9.499756555602935691705799805779, −8.998856828782080792625965019814, −8.037284899142455764759125072043, −7.04598163848948017495439506938, −5.54450183014960597903742450518, −4.61819611761172619370292157909, −3.37034421877396565540693327987, −1.48517922615807702866077263518,
0.72318971294014726046419862601, 2.09330994365511835033269380961, 3.57571208621274936039268884031, 5.51110293983403985803222925369, 6.44118221603083359049984824188, 7.50169149488329629351164359801, 8.182712498916013788561657975683, 8.817253749687301389657331984886, 9.985450854574256241010860144722, 10.70049880153713799999447312897