L(s) = 1 | + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−2.18 + 3.78i)5-s + (0.866 − 0.499i)6-s − 2.88i·7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (2.18 + 3.78i)10-s + (−3.19 − 0.900i)11-s − 0.999i·12-s + (−1.29 − 2.24i)13-s + (−2.50 − 1.44i)14-s + (−3.78 + 2.18i)15-s + (−0.5 + 0.866i)16-s + (1.84 + 1.06i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.976 + 1.69i)5-s + (0.353 − 0.204i)6-s − 1.09i·7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.690 + 1.19i)10-s + (−0.962 − 0.271i)11-s − 0.288i·12-s + (−0.360 − 0.623i)13-s + (−0.668 − 0.385i)14-s + (−0.976 + 0.563i)15-s + (−0.125 + 0.216i)16-s + (0.446 + 0.257i)17-s + ⋯ |
Λ(s)=(=(1254s/2ΓC(s)L(s)(−0.951+0.308i)Λ(2−s)
Λ(s)=(=(1254s/2ΓC(s+1/2)L(s)(−0.951+0.308i)Λ(1−s)
Degree: |
2 |
Conductor: |
1254
= 2⋅3⋅11⋅19
|
Sign: |
−0.951+0.308i
|
Analytic conductor: |
10.0132 |
Root analytic conductor: |
3.16437 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1254(901,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1254, ( :1/2), −0.951+0.308i)
|
Particular Values
L(1) |
≈ |
0.4980398113 |
L(21) |
≈ |
0.4980398113 |
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+(−0.866−0.5i)T |
| 11 | 1+(3.19+0.900i)T |
| 19 | 1+(3.70+2.30i)T |
good | 5 | 1+(2.18−3.78i)T+(−2.5−4.33i)T2 |
| 7 | 1+2.88iT−7T2 |
| 13 | 1+(1.29+2.24i)T+(−6.5+11.2i)T2 |
| 17 | 1+(−1.84−1.06i)T+(8.5+14.7i)T2 |
| 23 | 1+(−1.08−1.88i)T+(−11.5+19.9i)T2 |
| 29 | 1+(2.49+4.31i)T+(−14.5+25.1i)T2 |
| 31 | 1+7.35iT−31T2 |
| 37 | 1−4.74iT−37T2 |
| 41 | 1+(1.15−1.99i)T+(−20.5−35.5i)T2 |
| 43 | 1+(6.67+3.85i)T+(21.5+37.2i)T2 |
| 47 | 1+(1.55+2.69i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−1.98+1.14i)T+(26.5−45.8i)T2 |
| 59 | 1+(7.16+4.13i)T+(29.5+51.0i)T2 |
| 61 | 1+(11.9−6.89i)T+(30.5−52.8i)T2 |
| 67 | 1+(−3.45+1.99i)T+(33.5−58.0i)T2 |
| 71 | 1+(2.16+1.24i)T+(35.5+61.4i)T2 |
| 73 | 1+(−9.33−5.39i)T+(36.5+63.2i)T2 |
| 79 | 1+(−0.628+1.08i)T+(−39.5−68.4i)T2 |
| 83 | 1+15.2iT−83T2 |
| 89 | 1+(13.5−7.79i)T+(44.5−77.0i)T2 |
| 97 | 1+(6.65+3.84i)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
−9.824292863369524801159207973577, −8.273092653428186454729051981063, −7.70737728801213488140979807628, −7.03835163204500312507573996736, −6.00772113660564616107833125905, −4.66354120319327843055576217185, −3.80054867169156213284441106482, −3.17673399058683680884832321862, −2.36349158890872459047419404619, −0.16580089072450417861805258145,
1.72805871194295412115013906205, 3.10566753564374201734807486143, 4.27387160024016932820275145167, 5.00538697090743777708837157973, 5.63307493487385953862855546474, 6.93082270499200590087943181883, 7.80983426583572932651879124357, 8.383673198080292728090432666968, 8.915560470205306012548383323898, 9.588063753779319510162988596991