L(s) = 1 | + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−1.74 + 3.02i)5-s + (−0.5 + 0.866i)6-s + (2.04 − 3.53i)7-s − 0.999·8-s + 9-s + (1.74 + 3.02i)10-s + 4.08·11-s + (0.499 + 0.866i)12-s + (3.08 − 5.33i)13-s + (−2.04 − 3.53i)14-s + (1.74 − 3.02i)15-s + (−0.5 + 0.866i)16-s + (−3.83 − 6.64i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.780 + 1.35i)5-s + (−0.204 + 0.353i)6-s + (0.771 − 1.33i)7-s − 0.353·8-s + 0.333·9-s + (0.551 + 0.955i)10-s + 1.23·11-s + (0.144 + 0.249i)12-s + (0.854 − 1.47i)13-s + (−0.545 − 0.944i)14-s + (0.450 − 0.780i)15-s + (−0.125 + 0.216i)16-s + (−0.930 − 1.61i)17-s + ⋯ |
Λ(s)=(=(366s/2ΓC(s)L(s)(0.255+0.966i)Λ(2−s)
Λ(s)=(=(366s/2ΓC(s+1/2)L(s)(0.255+0.966i)Λ(1−s)
Degree: |
2 |
Conductor: |
366
= 2⋅3⋅61
|
Sign: |
0.255+0.966i
|
Analytic conductor: |
2.92252 |
Root analytic conductor: |
1.70953 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ366(13,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 366, ( :1/2), 0.255+0.966i)
|
Particular Values
L(1) |
≈ |
1.01394−0.780773i |
L(21) |
≈ |
1.01394−0.780773i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5+0.866i)T |
| 3 | 1+T |
| 61 | 1+(−1.88+7.57i)T |
good | 5 | 1+(1.74−3.02i)T+(−2.5−4.33i)T2 |
| 7 | 1+(−2.04+3.53i)T+(−3.5−6.06i)T2 |
| 11 | 1−4.08T+11T2 |
| 13 | 1+(−3.08+5.33i)T+(−6.5−11.2i)T2 |
| 17 | 1+(3.83+6.64i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−1−1.73i)T+(−9.5+16.4i)T2 |
| 23 | 1−1.59T+23T2 |
| 29 | 1+(−4.58−7.93i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−1−1.73i)T+(−15.5+26.8i)T2 |
| 37 | 1+2.59T+37T2 |
| 41 | 1−7.08T+41T2 |
| 43 | 1+(1.28−2.22i)T+(−21.5−37.2i)T2 |
| 47 | 1+(4.69+8.13i)T+(−23.5+40.7i)T2 |
| 53 | 1+6.30T+53T2 |
| 59 | 1+(−2.49+4.31i)T+(−29.5−51.0i)T2 |
| 67 | 1+(0.204−0.354i)T+(−33.5−58.0i)T2 |
| 71 | 1+(−5.08−8.79i)T+(−35.5+61.4i)T2 |
| 73 | 1+(−2.5−4.33i)T+(−36.5+63.2i)T2 |
| 79 | 1+(4.67−8.09i)T+(−39.5−68.4i)T2 |
| 83 | 1+(4.04−6.99i)T+(−41.5−71.8i)T2 |
| 89 | 1+15.8T+89T2 |
| 97 | 1+(−2.66−4.61i)T+(−48.5+84.0i)T2 |
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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
−11.06226236907451347449265835146, −10.83440129384430569778741599980, −9.848129090766527381068675983526, −8.315108618785523235353235887839, −7.15110907813220437291870109792, −6.64151957066115084756796554573, −5.07942057319333933364028445653, −3.97692576167411312892350600542, −3.14272014261601278187200170090, −0.985782425999355946798627676445,
1.61121903139439516655386133868, 4.19936781175274643038305336093, 4.52444475357406099646555038591, 5.85041306155827942672045660020, 6.53606426263852018214995564663, 8.065616019997167754424440484929, 8.776017942831535473935181055689, 9.213375654867571259403888807819, 11.25802714273247170837101090097, 11.72056815726889972161083978427