L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 3.48i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−1.74 − 3.02i)10-s + (2.32 − 1.34i)11-s + (−3.15 + 1.74i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.95 − 5.12i)17-s + (4.50 + 2.59i)19-s + (3.02 + 1.74i)20-s + (−1.34 + 2.32i)22-s + (3.52 + 6.10i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 1.55i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.551 − 0.955i)10-s + (0.700 − 0.404i)11-s + (−0.874 + 0.484i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.717 − 1.24i)17-s + (1.03 + 0.596i)19-s + (0.675 + 0.389i)20-s + (−0.285 + 0.495i)22-s + (0.735 + 1.27i)23-s + ⋯ |
Λ(s)=(=(1638s/2ΓC(s)L(s)(−0.473−0.880i)Λ(2−s)
Λ(s)=(=(1638s/2ΓC(s+1/2)L(s)(−0.473−0.880i)Λ(1−s)
Degree: |
2 |
Conductor: |
1638
= 2⋅32⋅7⋅13
|
Sign: |
−0.473−0.880i
|
Analytic conductor: |
13.0794 |
Root analytic conductor: |
3.61655 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1638(127,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1638, ( :1/2), −0.473−0.880i)
|
Particular Values
L(1) |
≈ |
1.300901409 |
L(21) |
≈ |
1.300901409 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.866−0.5i)T |
| 3 | 1 |
| 7 | 1+(−0.866−0.5i)T |
| 13 | 1+(3.15−1.74i)T |
good | 5 | 1−3.48iT−5T2 |
| 11 | 1+(−2.32+1.34i)T+(5.5−9.52i)T2 |
| 17 | 1+(−2.95+5.12i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−4.50−2.59i)T+(9.5+16.4i)T2 |
| 23 | 1+(−3.52−6.10i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−3.56−6.17i)T+(−14.5+25.1i)T2 |
| 31 | 1−0.782iT−31T2 |
| 37 | 1+(−6.76+3.90i)T+(18.5−32.0i)T2 |
| 41 | 1+(0.136−0.0788i)T+(20.5−35.5i)T2 |
| 43 | 1+(0.165−0.285i)T+(−21.5−37.2i)T2 |
| 47 | 1−1.60iT−47T2 |
| 53 | 1+3.92T+53T2 |
| 59 | 1+(8.26+4.77i)T+(29.5+51.0i)T2 |
| 61 | 1+(7.70−13.3i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−0.837+0.483i)T+(33.5−58.0i)T2 |
| 71 | 1+(3.62+2.09i)T+(35.5+61.4i)T2 |
| 73 | 1+15.0iT−73T2 |
| 79 | 1+0.293T+79T2 |
| 83 | 1−2.87iT−83T2 |
| 89 | 1+(4.51−2.60i)T+(44.5−77.0i)T2 |
| 97 | 1+(2.73+1.57i)T+(48.5+84.0i)T2 |
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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
−9.525102742815466528116759368825, −9.084583718519104710342126303960, −7.63558936013382788797138075922, −7.44589544266410787433849460596, −6.63867582166535453157116187292, −5.78684499884123563227566893026, −4.87856568786880463447939136909, −3.39127925123866402381072766280, −2.74883803919799337556115608951, −1.35678539728457361497009440933,
0.69917211380920880250532149126, 1.54754253100856410646313792759, 2.87202064354737723262237851712, 4.25075064277953914167896289138, 4.77643461800186061988156860404, 5.79132524959693989833754593616, 6.88432051802404109211866170578, 7.956841250660286134280746472270, 8.259814759773995758183005970564, 9.246428660932733963551999814314