L(s) = 1 | + (−1.29 − 1.15i)3-s + 1.58·5-s + (2.64 + 0.0963i)7-s + (0.349 + 2.97i)9-s + 1.58·11-s + (2.40 − 4.16i)13-s + (−2.05 − 1.82i)15-s + (−2.69 + 4.67i)17-s + (3.54 + 6.14i)19-s + (−3.31 − 3.16i)21-s − 0.300·23-s − 2.47·25-s + (2.97 − 4.25i)27-s + (4.13 + 7.16i)29-s + (−1.35 − 2.34i)31-s + ⋯ |
L(s) = 1 | + (−0.747 − 0.664i)3-s + 0.710·5-s + (0.999 + 0.0364i)7-s + (0.116 + 0.993i)9-s + 0.478·11-s + (0.667 − 1.15i)13-s + (−0.530 − 0.472i)15-s + (−0.654 + 1.13i)17-s + (0.814 + 1.41i)19-s + (−0.722 − 0.691i)21-s − 0.0626·23-s − 0.495·25-s + (0.572 − 0.819i)27-s + (0.768 + 1.33i)29-s + (−0.243 − 0.421i)31-s + ⋯ |
Λ(s)=(=(1008s/2ΓC(s)L(s)(0.945+0.325i)Λ(2−s)
Λ(s)=(=(1008s/2ΓC(s+1/2)L(s)(0.945+0.325i)Λ(1−s)
Degree: |
2 |
Conductor: |
1008
= 24⋅32⋅7
|
Sign: |
0.945+0.325i
|
Analytic conductor: |
8.04892 |
Root analytic conductor: |
2.83706 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1008(193,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1008, ( :1/2), 0.945+0.325i)
|
Particular Values
L(1) |
≈ |
1.675755737 |
L(21) |
≈ |
1.675755737 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(1.29+1.15i)T |
| 7 | 1+(−2.64−0.0963i)T |
good | 5 | 1−1.58T+5T2 |
| 11 | 1−1.58T+11T2 |
| 13 | 1+(−2.40+4.16i)T+(−6.5−11.2i)T2 |
| 17 | 1+(2.69−4.67i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−3.54−6.14i)T+(−9.5+16.4i)T2 |
| 23 | 1+0.300T+23T2 |
| 29 | 1+(−4.13−7.16i)T+(−14.5+25.1i)T2 |
| 31 | 1+(1.35+2.34i)T+(−15.5+26.8i)T2 |
| 37 | 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−2.93+5.08i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−0.833−1.44i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−1.33+2.30i)T+(−23.5−40.7i)T2 |
| 53 | 1+(−2.44+4.23i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−3.23−5.60i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−2.23+3.87i)T+(−30.5−52.8i)T2 |
| 67 | 1+(5.02+8.70i)T+(−33.5+58.0i)T2 |
| 71 | 1+12.7T+71T2 |
| 73 | 1+(−8.02+13.9i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−4.19+7.26i)T+(−39.5−68.4i)T2 |
| 83 | 1+(1.18+2.04i)T+(−41.5+71.8i)T2 |
| 89 | 1+(−1.60−2.78i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−0.712−1.23i)T+(−48.5+84.0i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.34172452314358318725386424921, −8.952685463622286744520086635429, −8.122984171972419401304774348328, −7.48501929437643842840412554001, −6.24518668740519708190786609549, −5.79069348042530900973526298359, −4.95514572151598576087960067765, −3.67426835935735837502067876534, −2.01720711100601881349038351943, −1.22166088279251945802949777836,
1.09739730595385598535311266963, 2.51252440684941007060492351017, 4.12178166477437790324780144130, 4.73031098073847314139868641832, 5.62744788924726695899493397715, 6.51871537888266964889641056207, 7.27115012664700137152391317463, 8.675432962954031932116098866256, 9.292390092876109516648246530816, 9.910140797316017713160943823936