L(s) = 1 | + (0.382 + 1.36i)2-s + (−1.70 + 1.04i)4-s − 2.05i·5-s − 4.18i·7-s + (−2.07 − 1.92i)8-s + (2.79 − 0.785i)10-s − 11-s + 3.27·13-s + (5.70 − 1.60i)14-s + (1.83 − 3.55i)16-s + 0.332i·17-s − 8.47i·19-s + (2.13 + 3.50i)20-s + (−0.382 − 1.36i)22-s − 3.52·23-s + ⋯ |
L(s) = 1 | + (0.270 + 0.962i)2-s + (−0.853 + 0.520i)4-s − 0.918i·5-s − 1.58i·7-s + (−0.732 − 0.681i)8-s + (0.884 − 0.248i)10-s − 0.301·11-s + 0.908·13-s + (1.52 − 0.428i)14-s + (0.457 − 0.888i)16-s + 0.0806i·17-s − 1.94i·19-s + (0.478 + 0.784i)20-s + (−0.0815 − 0.290i)22-s − 0.734·23-s + ⋯ |
Λ(s)=(=(396s/2ΓC(s)L(s)(0.918+0.396i)Λ(2−s)
Λ(s)=(=(396s/2ΓC(s+1/2)L(s)(0.918+0.396i)Λ(1−s)
Degree: |
2 |
Conductor: |
396
= 22⋅32⋅11
|
Sign: |
0.918+0.396i
|
Analytic conductor: |
3.16207 |
Root analytic conductor: |
1.77822 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ396(287,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 396, ( :1/2), 0.918+0.396i)
|
Particular Values
L(1) |
≈ |
1.25607−0.259687i |
L(21) |
≈ |
1.25607−0.259687i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.382−1.36i)T |
| 3 | 1 |
| 11 | 1+T |
good | 5 | 1+2.05iT−5T2 |
| 7 | 1+4.18iT−7T2 |
| 13 | 1−3.27T+13T2 |
| 17 | 1−0.332iT−17T2 |
| 19 | 1+8.47iT−19T2 |
| 23 | 1+3.52T+23T2 |
| 29 | 1−9.52iT−29T2 |
| 31 | 1−3.95iT−31T2 |
| 37 | 1+1.37T+37T2 |
| 41 | 1+5.42iT−41T2 |
| 43 | 1−1.53iT−43T2 |
| 47 | 1−6.93T+47T2 |
| 53 | 1+8.83iT−53T2 |
| 59 | 1+3.70T+59T2 |
| 61 | 1−14.8T+61T2 |
| 67 | 1+6.32iT−67T2 |
| 71 | 1+5.77T+71T2 |
| 73 | 1+8.72T+73T2 |
| 79 | 1−13.2iT−79T2 |
| 83 | 1−0.984T+83T2 |
| 89 | 1−1.46iT−89T2 |
| 97 | 1−5.99T+97T2 |
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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
−11.15790097560729718369442655977, −10.26726716472901759902679923456, −9.038403719142838851045104074052, −8.456384781330463447822761718325, −7.30880442000563854247396679078, −6.69988466345119947933842410845, −5.28838400329463744523290586371, −4.51273456134865882883886316876, −3.50885598590578174310988645262, −0.821147894588993495330444487543,
2.00402129091665141165057498006, 2.97978073604372772667979378369, 4.12137723581635910225304670811, 5.77367236500695092818379237051, 6.04868495890675277940577105990, 7.925093762678555068795812846151, 8.738950789420074186129728018769, 9.803609639227476326457623421102, 10.47963969912118046570300426398, 11.53185583232925167227952263659