L(s) = 1 | + (0.976 + 1.02i)2-s + (0.268 + 0.182i)3-s + (−0.0944 + 1.99i)4-s + (3.60 + 0.269i)5-s + (0.0746 + 0.453i)6-s + (−1.91 − 1.82i)7-s + (−2.13 + 1.85i)8-s + (−1.05 − 2.69i)9-s + (3.23 + 3.94i)10-s + (0.946 + 0.371i)11-s + (−0.390 + 0.518i)12-s + (−2.72 + 2.17i)13-s + (−0.00528 − 3.74i)14-s + (0.916 + 0.731i)15-s + (−3.98 − 0.377i)16-s + (−1.83 − 1.97i)17-s + ⋯ |
L(s) = 1 | + (0.690 + 0.723i)2-s + (0.154 + 0.105i)3-s + (−0.0472 + 0.998i)4-s + (1.61 + 0.120i)5-s + (0.0304 + 0.184i)6-s + (−0.724 − 0.689i)7-s + (−0.755 + 0.655i)8-s + (−0.352 − 0.898i)9-s + (1.02 + 1.24i)10-s + (0.285 + 0.111i)11-s + (−0.112 + 0.149i)12-s + (−0.755 + 0.602i)13-s + (−0.00141 − 0.999i)14-s + (0.236 + 0.188i)15-s + (−0.995 − 0.0943i)16-s + (−0.444 − 0.478i)17-s + ⋯ |
Λ(s)=(=(196s/2ΓC(s)L(s)(0.450−0.892i)Λ(2−s)
Λ(s)=(=(196s/2ΓC(s+1/2)L(s)(0.450−0.892i)Λ(1−s)
Degree: |
2 |
Conductor: |
196
= 22⋅72
|
Sign: |
0.450−0.892i
|
Analytic conductor: |
1.56506 |
Root analytic conductor: |
1.25102 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ196(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 196, ( :1/2), 0.450−0.892i)
|
Particular Values
L(1) |
≈ |
1.60791+0.989640i |
L(21) |
≈ |
1.60791+0.989640i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.976−1.02i)T |
| 7 | 1+(1.91+1.82i)T |
good | 3 | 1+(−0.268−0.182i)T+(1.09+2.79i)T2 |
| 5 | 1+(−3.60−0.269i)T+(4.94+0.745i)T2 |
| 11 | 1+(−0.946−0.371i)T+(8.06+7.48i)T2 |
| 13 | 1+(2.72−2.17i)T+(2.89−12.6i)T2 |
| 17 | 1+(1.83+1.97i)T+(−1.27+16.9i)T2 |
| 19 | 1+(2.18−3.77i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−4.02+4.33i)T+(−1.71−22.9i)T2 |
| 29 | 1+(−1.26−5.55i)T+(−26.1+12.5i)T2 |
| 31 | 1+(2.93+5.08i)T+(−15.5+26.8i)T2 |
| 37 | 1+(2.33+0.721i)T+(30.5+20.8i)T2 |
| 41 | 1+(2.01+4.17i)T+(−25.5+32.0i)T2 |
| 43 | 1+(−1.13+2.35i)T+(−26.8−33.6i)T2 |
| 47 | 1+(−1.76+0.266i)T+(44.9−13.8i)T2 |
| 53 | 1+(−7.54+2.32i)T+(43.7−29.8i)T2 |
| 59 | 1+(−1.04−13.8i)T+(−58.3+8.79i)T2 |
| 61 | 1+(1.60−5.20i)T+(−50.4−34.3i)T2 |
| 67 | 1+(−12.0+6.98i)T+(33.5−58.0i)T2 |
| 71 | 1+(6.12+1.39i)T+(63.9+30.8i)T2 |
| 73 | 1+(−0.391+2.59i)T+(−69.7−21.5i)T2 |
| 79 | 1+(−8.77−5.06i)T+(39.5+68.4i)T2 |
| 83 | 1+(9.43−11.8i)T+(−18.4−80.9i)T2 |
| 89 | 1+(3.57−1.40i)T+(65.2−60.5i)T2 |
| 97 | 1−13.8iT−97T2 |
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
−12.90605551865408042486213457427, −12.11311132598404250569090002429, −10.57155845436452298637546452544, −9.485837337230696867553608952294, −8.867486137738589577053088167096, −7.01739435734067323248781915406, −6.49953154186960013313300943659, −5.44097897469576626846341071406, −4.00640824814492293688146882031, −2.57351785610940353612962619274,
2.02866093142643035613051914594, 2.92822293000588688095801486217, 4.99142988789807192240941241977, 5.72802583020417240798556614232, 6.73465515125069556027795424107, 8.748366573622457618811712970633, 9.578489364491978753162043109759, 10.35130198947635444540001396009, 11.36742591424558317096897097778, 12.71818857668034559558842990233