L(s) = 1 | + (−1.32 + 2.29i)2-s + (−2.5 − 4.33i)4-s + 7.93·8-s + (2.64 + 4.58i)11-s + (−5.49 + 9.52i)16-s − 14·22-s + (−2.64 + 4.58i)23-s + (2.5 + 4.33i)25-s − 10.5·29-s + (−6.61 − 11.4i)32-s + (−3 + 5.19i)37-s + 12·43-s + (13.2 − 22.9i)44-s + (−7 − 12.1i)46-s − 13.2·50-s + ⋯ |
L(s) = 1 | + (−0.935 + 1.62i)2-s + (−1.25 − 2.16i)4-s + 2.80·8-s + (0.797 + 1.38i)11-s + (−1.37 + 2.38i)16-s − 2.98·22-s + (−0.551 + 0.955i)23-s + (0.5 + 0.866i)25-s − 1.96·29-s + (−1.16 − 2.02i)32-s + (−0.493 + 0.854i)37-s + 1.82·43-s + (1.99 − 3.45i)44-s + (−1.03 − 1.78i)46-s − 1.87·50-s + ⋯ |
Λ(s)=(=(441s/2ΓC(s)L(s)(−0.991−0.126i)Λ(2−s)
Λ(s)=(=(441s/2ΓC(s+1/2)L(s)(−0.991−0.126i)Λ(1−s)
Degree: |
2 |
Conductor: |
441
= 32⋅72
|
Sign: |
−0.991−0.126i
|
Analytic conductor: |
3.52140 |
Root analytic conductor: |
1.87654 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ441(226,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 441, ( :1/2), −0.991−0.126i)
|
Particular Values
L(1) |
≈ |
0.0431640+0.680175i |
L(21) |
≈ |
0.0431640+0.680175i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
good | 2 | 1+(1.32−2.29i)T+(−1−1.73i)T2 |
| 5 | 1+(−2.5−4.33i)T2 |
| 11 | 1+(−2.64−4.58i)T+(−5.5+9.52i)T2 |
| 13 | 1+13T2 |
| 17 | 1+(−8.5+14.7i)T2 |
| 19 | 1+(−9.5−16.4i)T2 |
| 23 | 1+(2.64−4.58i)T+(−11.5−19.9i)T2 |
| 29 | 1+10.5T+29T2 |
| 31 | 1+(−15.5+26.8i)T2 |
| 37 | 1+(3−5.19i)T+(−18.5−32.0i)T2 |
| 41 | 1+41T2 |
| 43 | 1−12T+43T2 |
| 47 | 1+(−23.5−40.7i)T2 |
| 53 | 1+(−5.29−9.16i)T+(−26.5+45.8i)T2 |
| 59 | 1+(−29.5+51.0i)T2 |
| 61 | 1+(−30.5−52.8i)T2 |
| 67 | 1+(2+3.46i)T+(−33.5+58.0i)T2 |
| 71 | 1+5.29T+71T2 |
| 73 | 1+(−36.5+63.2i)T2 |
| 79 | 1+(4−6.92i)T+(−39.5−68.4i)T2 |
| 83 | 1+83T2 |
| 89 | 1+(−44.5−77.0i)T2 |
| 97 | 1+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
−11.30181091589250696727956659177, −10.16521409613907406703997374738, −9.412375924765196334903300457434, −8.892251051294420503655452830306, −7.53675227555732028248516407599, −7.26599583516057262854233745818, −6.15070334950467214322174731646, −5.25115218957661253206508273131, −4.10954644861899663574982754642, −1.57524628942162217482476555325,
0.62738357114098857897071554810, 2.14009531752356935003789463378, 3.38024253899097073966283710723, 4.23004013721510196810552384000, 5.92317704149626451149910777062, 7.37985377558115264145451324182, 8.516195764939302457246275916807, 8.943031227210958481236175773362, 9.911812943306868928725525944038, 10.82609173296786453330035239923