L(s) = 1 | + (1.71 − 1.71i)2-s + (−3.25 − 3.25i)3-s − 1.86i·4-s + (−4.79 − 1.42i)5-s − 11.1·6-s + (2.88 − 2.88i)7-s + (3.66 + 3.66i)8-s + 12.1i·9-s + (−10.6 + 5.76i)10-s − 12.0·11-s + (−6.05 + 6.05i)12-s + (−12.4 − 12.4i)13-s − 9.86i·14-s + (10.9 + 20.2i)15-s + 19.9·16-s + (21.2 − 21.2i)17-s + ⋯ |
L(s) = 1 | + (0.855 − 0.855i)2-s + (−1.08 − 1.08i)3-s − 0.465i·4-s + (−0.958 − 0.284i)5-s − 1.85·6-s + (0.411 − 0.411i)7-s + (0.457 + 0.457i)8-s + 1.35i·9-s + (−1.06 + 0.576i)10-s − 1.09·11-s + (−0.504 + 0.504i)12-s + (−0.961 − 0.961i)13-s − 0.704i·14-s + (0.730 + 1.34i)15-s + 1.24·16-s + (1.25 − 1.25i)17-s + ⋯ |
Λ(s)=(=(115s/2ΓC(s)L(s)(−0.998−0.0569i)Λ(3−s)
Λ(s)=(=(115s/2ΓC(s+1)L(s)(−0.998−0.0569i)Λ(1−s)
Degree: |
2 |
Conductor: |
115
= 5⋅23
|
Sign: |
−0.998−0.0569i
|
Analytic conductor: |
3.13352 |
Root analytic conductor: |
1.77017 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ115(93,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 115, ( :1), −0.998−0.0569i)
|
Particular Values
L(23) |
≈ |
0.0306668+1.07564i |
L(21) |
≈ |
0.0306668+1.07564i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(4.79+1.42i)T |
| 23 | 1+(−3.39−3.39i)T |
good | 2 | 1+(−1.71+1.71i)T−4iT2 |
| 3 | 1+(3.25+3.25i)T+9iT2 |
| 7 | 1+(−2.88+2.88i)T−49iT2 |
| 11 | 1+12.0T+121T2 |
| 13 | 1+(12.4+12.4i)T+169iT2 |
| 17 | 1+(−21.2+21.2i)T−289iT2 |
| 19 | 1+8.56iT−361T2 |
| 29 | 1+41.0iT−841T2 |
| 31 | 1−6.48T+961T2 |
| 37 | 1+(−14.3+14.3i)T−1.36e3iT2 |
| 41 | 1+26.2T+1.68e3T2 |
| 43 | 1+(−41.3−41.3i)T+1.84e3iT2 |
| 47 | 1+(12.4−12.4i)T−2.20e3iT2 |
| 53 | 1+(4.51+4.51i)T+2.80e3iT2 |
| 59 | 1−48.3iT−3.48e3T2 |
| 61 | 1−89.1T+3.72e3T2 |
| 67 | 1+(−41.9+41.9i)T−4.48e3iT2 |
| 71 | 1+100.T+5.04e3T2 |
| 73 | 1+(−34.7−34.7i)T+5.32e3iT2 |
| 79 | 1+69.4iT−6.24e3T2 |
| 83 | 1+(81.6+81.6i)T+6.88e3iT2 |
| 89 | 1+53.2iT−7.92e3T2 |
| 97 | 1+(−106.+106.i)T−9.40e3iT2 |
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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
−12.68895079640322055203874456891, −11.82913021406457126417058687983, −11.33594029292396477849441077133, −10.24144805800172497586006972620, −7.68221950168710697718364292320, −7.61345431735882963927360327378, −5.48859063212663921384592409759, −4.69794216978951377198003136403, −2.82324552942026980066946616655, −0.68738082524536044905615791999,
3.79994141451987811696053481097, 4.88614326756161286926567983146, 5.57519877885699709128609156776, 6.93251566997441627737618155896, 8.141413460673175631741166491100, 10.01786906239878593570674259642, 10.72845019238742096714010288514, 11.88351861789290505926340165833, 12.65798012061106800426282249355, 14.41891503163859693889837784876