L(s) = 1 | + (1 + 1.73i)5-s + (1.5 + 2.59i)9-s + (2 − 3.46i)11-s − 2·13-s + (−3 + 5.19i)17-s + (4 + 6.92i)19-s + (0.500 − 0.866i)25-s + 6·29-s + (4 − 6.92i)31-s + (1 + 1.73i)37-s − 2·41-s − 4·43-s + (−3 + 5.19i)45-s + (−4 − 6.92i)47-s + (−3 + 5.19i)53-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.5 + 0.866i)9-s + (0.603 − 1.04i)11-s − 0.554·13-s + (−0.727 + 1.26i)17-s + (0.917 + 1.58i)19-s + (0.100 − 0.173i)25-s + 1.11·29-s + (0.718 − 1.24i)31-s + (0.164 + 0.284i)37-s − 0.312·41-s − 0.609·43-s + (−0.447 + 0.774i)45-s + (−0.583 − 1.01i)47-s + (−0.412 + 0.713i)53-s + ⋯ |
Λ(s)=(=(392s/2ΓC(s)L(s)(0.701−0.712i)Λ(2−s)
Λ(s)=(=(392s/2ΓC(s+1/2)L(s)(0.701−0.712i)Λ(1−s)
Degree: |
2 |
Conductor: |
392
= 23⋅72
|
Sign: |
0.701−0.712i
|
Analytic conductor: |
3.13013 |
Root analytic conductor: |
1.76921 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ392(361,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 392, ( :1/2), 0.701−0.712i)
|
Particular Values
L(1) |
≈ |
1.38639+0.581016i |
L(21) |
≈ |
1.38639+0.581016i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
good | 3 | 1+(−1.5−2.59i)T2 |
| 5 | 1+(−1−1.73i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−2+3.46i)T+(−5.5−9.52i)T2 |
| 13 | 1+2T+13T2 |
| 17 | 1+(3−5.19i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−4−6.92i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−11.5+19.9i)T2 |
| 29 | 1−6T+29T2 |
| 31 | 1+(−4+6.92i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−1−1.73i)T+(−18.5+32.0i)T2 |
| 41 | 1+2T+41T2 |
| 43 | 1+4T+43T2 |
| 47 | 1+(4+6.92i)T+(−23.5+40.7i)T2 |
| 53 | 1+(3−5.19i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−29.5−51.0i)T2 |
| 61 | 1+(3+5.19i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−2+3.46i)T+(−33.5−58.0i)T2 |
| 71 | 1+8T+71T2 |
| 73 | 1+(−5+8.66i)T+(−36.5−63.2i)T2 |
| 79 | 1+(8+13.8i)T+(−39.5+68.4i)T2 |
| 83 | 1+8T+83T2 |
| 89 | 1+(3+5.19i)T+(−44.5+77.0i)T2 |
| 97 | 1−6T+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
−11.32043421468784366133231299063, −10.36785676688309124009573448310, −9.929124609426543570709263219077, −8.539230659751575763907655294321, −7.76044616594718079628312705115, −6.54780586936613037623635497675, −5.86187912253282319375876365814, −4.46263616498577605798171719832, −3.19527635429439106026589972181, −1.80138803823286742340519672064,
1.15134921379481204045332394617, 2.82041761037243572874998032881, 4.52405899190915212718345758041, 5.06525128929975497774589015480, 6.71110751894114905772863100706, 7.14589684481559570253666464729, 8.725288688985365401154026504464, 9.439663427729552651315334271770, 9.917274417527311841030956296240, 11.40344511286972910858701886830