L(s) = 1 | + (−2.15 + 0.578i)2-s + (0.866 − 0.5i)4-s + (3.31 − 3.74i)5-s + (6.83 − 1.83i)7-s + (4.74 − 4.74i)8-s + (−5 + 10i)10-s + (7.90 − 13.6i)11-s + (−13.6 − 3.66i)13-s + (−13.6 + 7.90i)14-s + (−9.49 + 16.4i)16-s + (−3.16 − 3.16i)17-s + 18i·19-s + (1.00 − 4.89i)20-s + (−9.15 + 34.1i)22-s + (4.31 + 1.15i)23-s + ⋯ |
L(s) = 1 | + (−1.07 + 0.289i)2-s + (0.216 − 0.125i)4-s + (0.663 − 0.748i)5-s + (0.975 − 0.261i)7-s + (0.592 − 0.592i)8-s + (−0.5 + i)10-s + (0.718 − 1.24i)11-s + (−1.05 − 0.281i)13-s + (−0.978 + 0.564i)14-s + (−0.593 + 1.02i)16-s + (−0.186 − 0.186i)17-s + 0.947i·19-s + (0.0501 − 0.244i)20-s + (−0.415 + 1.55i)22-s + (0.187 + 0.0503i)23-s + ⋯ |
Λ(s)=(=(405s/2ΓC(s)L(s)(0.176+0.984i)Λ(3−s)
Λ(s)=(=(405s/2ΓC(s+1)L(s)(0.176+0.984i)Λ(1−s)
Degree: |
2 |
Conductor: |
405
= 34⋅5
|
Sign: |
0.176+0.984i
|
Analytic conductor: |
11.0354 |
Root analytic conductor: |
3.32196 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ405(217,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 405, ( :1), 0.176+0.984i)
|
Particular Values
L(23) |
≈ |
0.733525−0.613839i |
L(21) |
≈ |
0.733525−0.613839i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1+(−3.31+3.74i)T |
good | 2 | 1+(2.15−0.578i)T+(3.46−2i)T2 |
| 7 | 1+(−6.83+1.83i)T+(42.4−24.5i)T2 |
| 11 | 1+(−7.90+13.6i)T+(−60.5−104.i)T2 |
| 13 | 1+(13.6+3.66i)T+(146.+84.5i)T2 |
| 17 | 1+(3.16+3.16i)T+289iT2 |
| 19 | 1−18iT−361T2 |
| 23 | 1+(−4.31−1.15i)T+(458.+264.5i)T2 |
| 29 | 1+(41.0+23.7i)T+(420.5+728.i)T2 |
| 31 | 1+(4+6.92i)T+(−480.5+832.i)T2 |
| 37 | 1+(−10−10i)T+1.36e3iT2 |
| 41 | 1+(15.8+27.3i)T+(−840.5+1.45e3i)T2 |
| 43 | 1+(−3.66−13.6i)T+(−1.60e3+924.5i)T2 |
| 47 | 1+(−56.1+15.0i)T+(1.91e3−1.10e3i)T2 |
| 53 | 1+(−25.2+25.2i)T−2.80e3iT2 |
| 59 | 1+(41.0−23.7i)T+(1.74e3−3.01e3i)T2 |
| 61 | 1+(−29+50.2i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(−25.6+95.6i)T+(−3.88e3−2.24e3i)T2 |
| 71 | 1+63.2T+5.04e3T2 |
| 73 | 1+(−55+55i)T−5.32e3iT2 |
| 79 | 1+(10.3+6i)T+(3.12e3+5.40e3i)T2 |
| 83 | 1+(−19.6−73.4i)T+(−5.96e3+3.44e3i)T2 |
| 89 | 1−7.92e3T2 |
| 97 | 1+(−6.83+1.83i)T+(8.14e3−4.70e3i)T2 |
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
−10.59627136880629560482243843667, −9.665158645539093076916396473636, −9.001803588895540276673440069570, −8.176074845572861318916431336845, −7.50222469902229662242805786390, −6.15432915649915668382502163888, −5.08754255937494039780055974040, −3.93024760374787806435654056782, −1.84061328904944644722237138066, −0.62317473013916487506518725824,
1.60851625053961948931938487570, 2.41338314048273461268797627623, 4.46681186747022836630959840992, 5.39336920365767152210646730366, 6.98880972242801253458531872429, 7.48376854104870079695094137878, 8.842042476356393778363611841104, 9.427898970200885379536157370781, 10.19040000567315746984462102111, 11.05939406930642730020779109301