L(s) = 1 | − 4.30i·2-s + 19.9i·3-s + 13.4·4-s + 85.7·6-s − 125. i·7-s − 195. i·8-s − 153.·9-s − 709.·11-s + 267. i·12-s + 169i·13-s − 541.·14-s − 412.·16-s + 1.92e3i·17-s + 660. i·18-s + 2.16e3·19-s + ⋯ |
L(s) = 1 | − 0.761i·2-s + 1.27i·3-s + 0.420·4-s + 0.972·6-s − 0.970i·7-s − 1.08i·8-s − 0.631·9-s − 1.76·11-s + 0.537i·12-s + 0.277i·13-s − 0.738·14-s − 0.402·16-s + 1.61i·17-s + 0.480i·18-s + 1.37·19-s + ⋯ |
Λ(s)=(=(325s/2ΓC(s)L(s)(0.447−0.894i)Λ(6−s)
Λ(s)=(=(325s/2ΓC(s+5/2)L(s)(0.447−0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
325
= 52⋅13
|
Sign: |
0.447−0.894i
|
Analytic conductor: |
52.1247 |
Root analytic conductor: |
7.21974 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ325(274,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 325, ( :5/2), 0.447−0.894i)
|
Particular Values
L(3) |
≈ |
1.868762296 |
L(21) |
≈ |
1.868762296 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 13 | 1−169iT |
good | 2 | 1+4.30iT−32T2 |
| 3 | 1−19.9iT−243T2 |
| 7 | 1+125.iT−1.68e4T2 |
| 11 | 1+709.T+1.61e5T2 |
| 17 | 1−1.92e3iT−1.41e6T2 |
| 19 | 1−2.16e3T+2.47e6T2 |
| 23 | 1+305.iT−6.43e6T2 |
| 29 | 1+4.51e3T+2.05e7T2 |
| 31 | 1−4.07e3T+2.86e7T2 |
| 37 | 1−1.23e4iT−6.93e7T2 |
| 41 | 1−1.44e4T+1.15e8T2 |
| 43 | 1−1.71e4iT−1.47e8T2 |
| 47 | 1−1.83e4iT−2.29e8T2 |
| 53 | 1+1.19e4iT−4.18e8T2 |
| 59 | 1−3.98e4T+7.14e8T2 |
| 61 | 1−4.49e3T+8.44e8T2 |
| 67 | 1−3.04e4iT−1.35e9T2 |
| 71 | 1+1.45e4T+1.80e9T2 |
| 73 | 1−5.90e4iT−2.07e9T2 |
| 79 | 1−4.33e4T+3.07e9T2 |
| 83 | 1+8.38e4iT−3.93e9T2 |
| 89 | 1+1.24e5T+5.58e9T2 |
| 97 | 1+9.38e4iT−8.58e9T2 |
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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
−10.81758068457538286230131278248, −10.11529049216487537033453336485, −9.717475237158655738298264330070, −8.143820951737986609468676762708, −7.22818184967704845702521039754, −5.81952955395064610627862380348, −4.59649190875203365213043732372, −3.70396176029634519911912063174, −2.74754488050629105512990070761, −1.18182845500242653281802114521,
0.49713505281481723689920794454, 2.19474872970745452066655912958, 2.76928972504518523842207236535, 5.36993818307635333805859072943, 5.60707924253176589405885651887, 7.05496998049196721546709166870, 7.50073226854556392182206880184, 8.238539906101703305680510650233, 9.447609336972541924403123672402, 10.80989694327476762588141881127