L(s) = 1 | + (−3 − 1.73i)2-s + (3.5 − 6.06i)3-s + (2 + 3.46i)4-s + 13.8i·5-s + (−21 + 12.1i)6-s + (19.5 − 11.2i)7-s + 13.8i·8-s + (−11 − 19.0i)9-s + (23.9 − 41.5i)10-s + (−19.5 − 11.2i)11-s + 28.0·12-s + (−13 + 45.0i)13-s − 78·14-s + (84 + 48.4i)15-s + (39.9 − 69.2i)16-s + (−13.5 − 23.3i)17-s + ⋯ |
L(s) = 1 | + (−1.06 − 0.612i)2-s + (0.673 − 1.16i)3-s + (0.250 + 0.433i)4-s + 1.23i·5-s + (−1.42 + 0.824i)6-s + (1.05 − 0.607i)7-s + 0.612i·8-s + (−0.407 − 0.705i)9-s + (0.758 − 1.31i)10-s + (−0.534 − 0.308i)11-s + 0.673·12-s + (−0.277 + 0.960i)13-s − 1.48·14-s + (1.44 + 0.834i)15-s + (0.624 − 1.08i)16-s + (−0.192 − 0.333i)17-s + ⋯ |
Λ(s)=(=(13s/2ΓC(s)L(s)(0.252+0.967i)Λ(4−s)
Λ(s)=(=(13s/2ΓC(s+3/2)L(s)(0.252+0.967i)Λ(1−s)
Degree: |
2 |
Conductor: |
13
|
Sign: |
0.252+0.967i
|
Analytic conductor: |
0.767024 |
Root analytic conductor: |
0.875799 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ13(4,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 13, ( :3/2), 0.252+0.967i)
|
Particular Values
L(2) |
≈ |
0.566359−0.437474i |
L(21) |
≈ |
0.566359−0.437474i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1+(13−45.0i)T |
good | 2 | 1+(3+1.73i)T+(4+6.92i)T2 |
| 3 | 1+(−3.5+6.06i)T+(−13.5−23.3i)T2 |
| 5 | 1−13.8iT−125T2 |
| 7 | 1+(−19.5+11.2i)T+(171.5−297.i)T2 |
| 11 | 1+(19.5+11.2i)T+(665.5+1.15e3i)T2 |
| 17 | 1+(13.5+23.3i)T+(−2.45e3+4.25e3i)T2 |
| 19 | 1+(76.5−44.1i)T+(3.42e3−5.94e3i)T2 |
| 23 | 1+(28.5−49.3i)T+(−6.08e3−1.05e4i)T2 |
| 29 | 1+(−34.5+59.7i)T+(−1.21e4−2.11e4i)T2 |
| 31 | 1+72.7iT−2.97e4T2 |
| 37 | 1+(34.5+19.9i)T+(2.53e4+4.38e4i)T2 |
| 41 | 1+(340.5+196.i)T+(3.44e4+5.96e4i)T2 |
| 43 | 1+(−42.5−73.6i)T+(−3.97e4+6.88e4i)T2 |
| 47 | 1+342.iT−1.03e5T2 |
| 53 | 1−426T+1.48e5T2 |
| 59 | 1+(16.5−9.52i)T+(1.02e5−1.77e5i)T2 |
| 61 | 1+(−8.5−14.7i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(−142.5−82.2i)T+(1.50e5+2.60e5i)T2 |
| 71 | 1+(−505.5+291.i)T+(1.78e5−3.09e5i)T2 |
| 73 | 1−1.00e3iT−3.89e5T2 |
| 79 | 1+1.24e3T+4.93e5T2 |
| 83 | 1+426.iT−5.71e5T2 |
| 89 | 1+(−265.5−153.i)T+(3.52e5+6.10e5i)T2 |
| 97 | 1+(−1.06e3+617.i)T+(4.56e5−7.90e5i)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
−18.85927168084700490843340412488, −18.44175080340776231630660574013, −17.26214822321643194606012520434, −14.59910339934377349528497014381, −13.78044923184104415161506893934, −11.57781508819803023179857827817, −10.39776468923365132375400991003, −8.360288387817286688492718603371, −7.18864249379947825917987191008, −2.11014574367122488708249141919,
4.80198786544075143735648168099, 8.231085512020065232240113341638, 8.868693986454882017884086778155, 10.30179298841582890889348698943, 12.71056452343602839582459843870, 14.96920099474815101760947911563, 15.75822802309119482862262646720, 16.99354970851497277558510659429, 18.07827698741142481588090997022, 19.88636005144530613870746315734