L(s) = 1 | + (0.623 + 1.20i)3-s + (−3.67 + 0.351i)5-s + (0.914 − 2.48i)7-s + (0.667 − 0.937i)9-s + (4.82 − 1.67i)11-s + (0.293 + 0.999i)13-s + (−2.71 − 4.22i)15-s + (6.56 + 2.62i)17-s + (−7.01 + 2.80i)19-s + (3.57 − 0.441i)21-s + (1.34 − 4.60i)23-s + (8.49 − 1.63i)25-s + (5.58 + 0.803i)27-s + (−0.133 − 0.928i)29-s + (3.61 − 0.171i)31-s + ⋯ |
L(s) = 1 | + (0.359 + 0.697i)3-s + (−1.64 + 0.157i)5-s + (0.345 − 0.938i)7-s + (0.222 − 0.312i)9-s + (1.45 − 0.503i)11-s + (0.0813 + 0.277i)13-s + (−0.701 − 1.09i)15-s + (1.59 + 0.637i)17-s + (−1.60 + 0.643i)19-s + (0.779 − 0.0962i)21-s + (0.280 − 0.959i)23-s + (1.69 − 0.327i)25-s + (1.07 + 0.154i)27-s + (−0.0248 − 0.172i)29-s + (0.648 − 0.0308i)31-s + ⋯ |
Λ(s)=(=(644s/2ΓC(s)L(s)(0.999−0.0381i)Λ(2−s)
Λ(s)=(=(644s/2ΓC(s+1/2)L(s)(0.999−0.0381i)Λ(1−s)
Degree: |
2 |
Conductor: |
644
= 22⋅7⋅23
|
Sign: |
0.999−0.0381i
|
Analytic conductor: |
5.14236 |
Root analytic conductor: |
2.26767 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ644(493,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 644, ( :1/2), 0.999−0.0381i)
|
Particular Values
L(1) |
≈ |
1.45867+0.0278501i |
L(21) |
≈ |
1.45867+0.0278501i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1+(−0.914+2.48i)T |
| 23 | 1+(−1.34+4.60i)T |
good | 3 | 1+(−0.623−1.20i)T+(−1.74+2.44i)T2 |
| 5 | 1+(3.67−0.351i)T+(4.90−0.946i)T2 |
| 11 | 1+(−4.82+1.67i)T+(8.64−6.79i)T2 |
| 13 | 1+(−0.293−0.999i)T+(−10.9+7.02i)T2 |
| 17 | 1+(−6.56−2.62i)T+(12.3+11.7i)T2 |
| 19 | 1+(7.01−2.80i)T+(13.7−13.1i)T2 |
| 29 | 1+(0.133+0.928i)T+(−27.8+8.17i)T2 |
| 31 | 1+(−3.61+0.171i)T+(30.8−2.94i)T2 |
| 37 | 1+(−5.13−3.65i)T+(12.1+34.9i)T2 |
| 41 | 1+(−2.40−1.10i)T+(26.8+30.9i)T2 |
| 43 | 1+(0.463−0.720i)T+(−17.8−39.1i)T2 |
| 47 | 1+(−10.5+6.07i)T+(23.5−40.7i)T2 |
| 53 | 1+(9.58+10.0i)T+(−2.52+52.9i)T2 |
| 59 | 1+(−5.07−1.23i)T+(52.4+27.0i)T2 |
| 61 | 1+(4.45+2.29i)T+(35.3+49.6i)T2 |
| 67 | 1+(−0.918−4.76i)T+(−62.2+24.9i)T2 |
| 71 | 1+(−4.77−5.50i)T+(−10.1+70.2i)T2 |
| 73 | 1+(5.52−7.02i)T+(−17.2−70.9i)T2 |
| 79 | 1+(2.59−2.71i)T+(−3.75−78.9i)T2 |
| 83 | 1+(4.28+9.38i)T+(−54.3+62.7i)T2 |
| 89 | 1+(−0.167+3.52i)T+(−88.5−8.45i)T2 |
| 97 | 1+(1.43−3.14i)T+(−63.5−73.3i)T2 |
show more | |
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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
−10.56565117357730516113452151974, −9.836760323232073724975961187479, −8.563610989706066208654287477524, −8.196062737830660523965463832205, −7.08872153649672852834471808119, −6.29045206062656194023059029011, −4.39709544339525686666365511997, −4.04170503912070406371034997658, −3.37188217782036158938994328123, −1.00782175138635689381929019595,
1.24233334118439210092835146587, 2.74105505077278310298522941914, 3.98915062717387536852587189922, 4.85076281125819160288511607715, 6.24422276225734339173319057897, 7.37889665907953086361889704559, 7.77811925599184868039609855398, 8.678450995270038765384247866248, 9.392634433859678890283688443078, 10.83442719678895424954580014020