L(s) = 1 | + (−0.285 + 0.493i)3-s + (12.7 + 47.3i)7-s + (40.3 + 69.8i)9-s + (−89.1 + 154. i)11-s − 116.·13-s + (−182. + 316. i)17-s + (98.9 − 57.1i)19-s + (−26.9 − 7.21i)21-s + (−38.6 + 22.3i)23-s − 92.1·27-s − 1.12e3·29-s + (−9.18 − 5.30i)31-s + (−50.8 − 88.1i)33-s + (1.30e3 − 756. i)37-s + (33.1 − 57.3i)39-s + ⋯ |
L(s) = 1 | + (−0.0316 + 0.0548i)3-s + (0.259 + 0.965i)7-s + (0.497 + 0.862i)9-s + (−0.737 + 1.27i)11-s − 0.687·13-s + (−0.631 + 1.09i)17-s + (0.274 − 0.158i)19-s + (−0.0612 − 0.0163i)21-s + (−0.0730 + 0.0421i)23-s − 0.126·27-s − 1.34·29-s + (−0.00955 − 0.00551i)31-s + (−0.0467 − 0.0809i)33-s + (0.956 − 0.552i)37-s + (0.0217 − 0.0377i)39-s + ⋯ |
Λ(s)=(=(700s/2ΓC(s)L(s)(−0.900+0.434i)Λ(5−s)
Λ(s)=(=(700s/2ΓC(s+2)L(s)(−0.900+0.434i)Λ(1−s)
Degree: |
2 |
Conductor: |
700
= 22⋅52⋅7
|
Sign: |
−0.900+0.434i
|
Analytic conductor: |
72.3589 |
Root analytic conductor: |
8.50640 |
Motivic weight: |
4 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ700(549,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 700, ( :2), −0.900+0.434i)
|
Particular Values
L(25) |
≈ |
0.7403180752 |
L(21) |
≈ |
0.7403180752 |
L(3) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 7 | 1+(−12.7−47.3i)T |
good | 3 | 1+(0.285−0.493i)T+(−40.5−70.1i)T2 |
| 11 | 1+(89.1−154.i)T+(−7.32e3−1.26e4i)T2 |
| 13 | 1+116.T+2.85e4T2 |
| 17 | 1+(182.−316.i)T+(−4.17e4−7.23e4i)T2 |
| 19 | 1+(−98.9+57.1i)T+(6.51e4−1.12e5i)T2 |
| 23 | 1+(38.6−22.3i)T+(1.39e5−2.42e5i)T2 |
| 29 | 1+1.12e3T+7.07e5T2 |
| 31 | 1+(9.18+5.30i)T+(4.61e5+7.99e5i)T2 |
| 37 | 1+(−1.30e3+756.i)T+(9.37e5−1.62e6i)T2 |
| 41 | 1+515.iT−2.82e6T2 |
| 43 | 1+2.07e3iT−3.41e6T2 |
| 47 | 1+(806.+1.39e3i)T+(−2.43e6+4.22e6i)T2 |
| 53 | 1+(−2.25e3−1.30e3i)T+(3.94e6+6.83e6i)T2 |
| 59 | 1+(−3.96e3−2.28e3i)T+(6.05e6+1.04e7i)T2 |
| 61 | 1+(−4.87e3+2.81e3i)T+(6.92e6−1.19e7i)T2 |
| 67 | 1+(3.19e3+1.84e3i)T+(1.00e7+1.74e7i)T2 |
| 71 | 1+813.T+2.54e7T2 |
| 73 | 1+(−2.33e3+4.04e3i)T+(−1.41e7−2.45e7i)T2 |
| 79 | 1+(633.+1.09e3i)T+(−1.94e7+3.37e7i)T2 |
| 83 | 1+1.06e4T+4.74e7T2 |
| 89 | 1+(8.33e3−4.81e3i)T+(3.13e7−5.43e7i)T2 |
| 97 | 1−9.56e3T+8.85e7T2 |
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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
−10.29445902573582754779733859768, −9.590188207009754299032491518615, −8.610517272901099714659248178014, −7.69762668144533430589466798759, −7.04738463370071899675206532602, −5.68188720515875183329082771957, −5.00248381227199869026504839307, −4.07434051684421468527621524692, −2.37756288013286022292427659397, −1.90472026725764601981861096549,
0.18388482165160633967992554773, 1.08694910725535467962997288193, 2.69301978798389561616227020540, 3.74336828286547567574742326469, 4.73357577675393325508745800077, 5.77854721376342499608644251390, 6.86449050897954644426697403875, 7.53064017133178218603608917140, 8.447794115424424204221421362234, 9.537614938553878979242131989830