L(s) = 1 | − i·2-s − 4-s − i·5-s − 4.60i·7-s + i·8-s − 10-s + 3.60·13-s − 4.60·14-s + 16-s − 4.60·17-s − 4.60i·19-s + i·20-s + 1.39·23-s − 25-s − 3.60i·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 1.74i·7-s + 0.353i·8-s − 0.316·10-s + 1.00·13-s − 1.23·14-s + 0.250·16-s − 1.11·17-s − 1.05i·19-s + 0.223i·20-s + 0.290·23-s − 0.200·25-s − 0.707i·26-s + ⋯ |
Λ(s)=(=(1170s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(1170s/2ΓC(s+1/2)L(s)−Λ(1−s)
Degree: |
2 |
Conductor: |
1170
= 2⋅32⋅5⋅13
|
Sign: |
−1
|
Analytic conductor: |
9.34249 |
Root analytic conductor: |
3.05654 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1170(181,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1170, ( :1/2), −1)
|
Particular Values
L(1) |
≈ |
1.170007064 |
L(21) |
≈ |
1.170007064 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+iT |
| 3 | 1 |
| 5 | 1+iT |
| 13 | 1−3.60T |
good | 7 | 1+4.60iT−7T2 |
| 11 | 1−11T2 |
| 17 | 1+4.60T+17T2 |
| 19 | 1+4.60iT−19T2 |
| 23 | 1−1.39T+23T2 |
| 29 | 1+4.60T+29T2 |
| 31 | 1−6iT−31T2 |
| 37 | 1+9.21iT−37T2 |
| 41 | 1+3.21iT−41T2 |
| 43 | 1−8T+43T2 |
| 47 | 1−9.21iT−47T2 |
| 53 | 1+6T+53T2 |
| 59 | 1−9.21iT−59T2 |
| 61 | 1+11.2T+61T2 |
| 67 | 1+3.21iT−67T2 |
| 71 | 1+9.21iT−71T2 |
| 73 | 1+1.39iT−73T2 |
| 79 | 1+14.4T+79T2 |
| 83 | 1+2.78iT−83T2 |
| 89 | 1+15.2iT−89T2 |
| 97 | 1−1.39iT−97T2 |
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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
−9.252645784952432429332199795045, −8.891767460786387990176091042533, −7.69396526152934103147073935464, −7.03274628397051416972650465303, −5.95308081148676960619149608294, −4.63297653432515527542396250964, −4.16042112515793033810378920628, −3.14775777238510614307352411497, −1.62545250391783214636567988570, −0.51487114833764607901905081384,
1.90099776105065766427192433005, 3.06447311149039710998548026074, 4.21200496971249717896478680118, 5.40331199792332387568937544010, 6.04304329254448087229724277265, 6.64149625038635540495788379015, 7.85416547768879583175197780591, 8.525544310527710369167190329911, 9.161138011573199076177604006441, 9.945491203669175515547786614445