L(s) = 1 | + (0.826 + 0.300i)2-s + (−0.592 − 1.62i)3-s + (−0.939 − 0.788i)4-s − 1.52i·6-s + (2.87 − 2.41i)7-s + (−1.41 − 2.45i)8-s + (−2.29 + 1.92i)9-s + (−0.180 − 1.02i)11-s + (−0.726 + 1.99i)12-s + (−2.99 + 1.08i)13-s + (3.10 − 1.13i)14-s + (−0.00727 − 0.0412i)16-s + (0.233 − 0.405i)17-s + (−2.47 + 0.902i)18-s + (−2.34 − 4.06i)19-s + ⋯ |
L(s) = 1 | + (0.584 + 0.212i)2-s + (−0.342 − 0.939i)3-s + (−0.469 − 0.394i)4-s − 0.621i·6-s + (1.08 − 0.913i)7-s + (−0.501 − 0.868i)8-s + (−0.766 + 0.642i)9-s + (−0.0545 − 0.309i)11-s + (−0.209 + 0.576i)12-s + (−0.830 + 0.302i)13-s + (0.830 − 0.302i)14-s + (−0.00181 − 0.0103i)16-s + (0.0567 − 0.0982i)17-s + (−0.584 + 0.212i)18-s + (−0.538 − 0.932i)19-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(−0.835+0.549i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(−0.835+0.549i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
−0.835+0.549i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(301,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), −0.835+0.549i)
|
Particular Values
L(1) |
≈ |
0.354564−1.18432i |
L(21) |
≈ |
0.354564−1.18432i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.592+1.62i)T |
| 5 | 1 |
good | 2 | 1+(−0.826−0.300i)T+(1.53+1.28i)T2 |
| 7 | 1+(−2.87+2.41i)T+(1.21−6.89i)T2 |
| 11 | 1+(0.180+1.02i)T+(−10.3+3.76i)T2 |
| 13 | 1+(2.99−1.08i)T+(9.95−8.35i)T2 |
| 17 | 1+(−0.233+0.405i)T+(−8.5−14.7i)T2 |
| 19 | 1+(2.34+4.06i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−4.11−3.45i)T+(3.99+22.6i)T2 |
| 29 | 1+(5.45+1.98i)T+(22.2+18.6i)T2 |
| 31 | 1+(3.14+2.63i)T+(5.38+30.5i)T2 |
| 37 | 1+(2.23−3.87i)T+(−18.5−32.0i)T2 |
| 41 | 1+(7.52−2.73i)T+(31.4−26.3i)T2 |
| 43 | 1+(2.11+11.9i)T+(−40.4+14.7i)T2 |
| 47 | 1+(−2.65+2.22i)T+(8.16−46.2i)T2 |
| 53 | 1−8.83T+53T2 |
| 59 | 1+(−2.36+13.4i)T+(−55.4−20.1i)T2 |
| 61 | 1+(−7.46+6.26i)T+(10.5−60.0i)T2 |
| 67 | 1+(−1.71+0.623i)T+(51.3−43.0i)T2 |
| 71 | 1+(3.85−6.67i)T+(−35.5−61.4i)T2 |
| 73 | 1+(0.407+0.705i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−3.81−1.38i)T+(60.5+50.7i)T2 |
| 83 | 1+(−15.9−5.81i)T+(63.5+53.3i)T2 |
| 89 | 1+(−5.19−9.00i)T+(−44.5+77.0i)T2 |
| 97 | 1+(1.06+6.02i)T+(−91.1+33.1i)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
−10.32571033903231957575504533795, −9.195713013298573971035362457778, −8.242100894859051867132825561355, −7.24235400752396300816685292314, −6.70542809336239920781890869585, −5.35485562563716158603113467226, −4.94934747095947924249752960568, −3.72529039646307801446107936313, −1.96743743901241239253370325181, −0.56867753597151268684574138849,
2.29294713796537206193624710606, 3.48430748220785210620397971537, 4.57355598820358008491044977073, 5.14516272183433376460249677986, 5.86965056779127845581559445539, 7.48025598289946419734313836695, 8.569564080621873707819527705566, 8.973754742740896181968551826176, 10.10013517463067973470823730292, 10.95043336631176467671598640072