L(s) = 1 | + (−1.58 − 1.58i)2-s + 1.00i·4-s + (−1.58 − 4.74i)5-s + (−5 − 5i)7-s + (−4.74 + 4.74i)8-s + (−5 + 10.0i)10-s + 15.8·11-s + (10 − 10i)13-s + 15.8i·14-s + 19·16-s + (3.16 + 3.16i)17-s + 18i·19-s + (4.74 − 1.58i)20-s + (−25 − 25i)22-s + (3.16 − 3.16i)23-s + ⋯ |
L(s) = 1 | + (−0.790 − 0.790i)2-s + 0.250i·4-s + (−0.316 − 0.948i)5-s + (−0.714 − 0.714i)7-s + (−0.592 + 0.592i)8-s + (−0.5 + 1.00i)10-s + 1.43·11-s + (0.769 − 0.769i)13-s + 1.12i·14-s + 1.18·16-s + (0.186 + 0.186i)17-s + 0.947i·19-s + (0.237 − 0.0790i)20-s + (−1.13 − 1.13i)22-s + (0.137 − 0.137i)23-s + ⋯ |
Λ(s)=(=(45s/2ΓC(s)L(s)(−0.640+0.767i)Λ(3−s)
Λ(s)=(=(45s/2ΓC(s+1)L(s)(−0.640+0.767i)Λ(1−s)
Degree: |
2 |
Conductor: |
45
= 32⋅5
|
Sign: |
−0.640+0.767i
|
Analytic conductor: |
1.22616 |
Root analytic conductor: |
1.10732 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ45(37,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 45, ( :1), −0.640+0.767i)
|
Particular Values
L(23) |
≈ |
0.282755−0.604270i |
L(21) |
≈ |
0.282755−0.604270i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1+(1.58+4.74i)T |
good | 2 | 1+(1.58+1.58i)T+4iT2 |
| 7 | 1+(5+5i)T+49iT2 |
| 11 | 1−15.8T+121T2 |
| 13 | 1+(−10+10i)T−169iT2 |
| 17 | 1+(−3.16−3.16i)T+289iT2 |
| 19 | 1−18iT−361T2 |
| 23 | 1+(−3.16+3.16i)T−529iT2 |
| 29 | 1+47.4iT−841T2 |
| 31 | 1−8T+961T2 |
| 37 | 1+(−10−10i)T+1.36e3iT2 |
| 41 | 1+31.6T+1.68e3T2 |
| 43 | 1+(−10+10i)T−1.84e3iT2 |
| 47 | 1+(−41.1−41.1i)T+2.20e3iT2 |
| 53 | 1+(25.2−25.2i)T−2.80e3iT2 |
| 59 | 1−47.4iT−3.48e3T2 |
| 61 | 1+58T+3.72e3T2 |
| 67 | 1+(−70−70i)T+4.48e3iT2 |
| 71 | 1−63.2T+5.04e3T2 |
| 73 | 1+(−55+55i)T−5.32e3iT2 |
| 79 | 1−12iT−6.24e3T2 |
| 83 | 1+(53.7−53.7i)T−6.88e3iT2 |
| 89 | 1−7.92e3T2 |
| 97 | 1+(5+5i)T+9.40e3iT2 |
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
−15.35778838387373975982491318513, −13.86588117448618704821133706944, −12.54584940612755552450589198395, −11.55497479671012218091792319421, −10.22841955442148026022444801860, −9.260270254761943393254256602495, −8.107359053703639145676963085828, −6.05875941023193682378623855277, −3.82324094743719083889806152585, −1.02551862370993760166493414529,
3.46164702436307439002314720361, 6.34599495110069946504508343750, 7.01301774144573481201652635605, 8.715102600631252747470021527276, 9.531570061267914645515259405950, 11.25567763653390900346230388590, 12.38260105783380688891419048359, 14.08158237580846356865275245363, 15.22377634246544515319830255750, 16.04267945303262888143365657873