L(s) = 1 | + (18.6 + 18.6i)2-s + (−274. − 663. i)3-s − 1.34e3i·4-s + (1.89e3 − 783. i)5-s + (7.26e3 − 1.75e4i)6-s + (−1.94e4 − 8.05e3i)7-s + (6.35e4 − 6.35e4i)8-s + (−2.39e5 + 2.39e5i)9-s + (5.00e4 + 2.07e4i)10-s + (−2.12e5 + 5.13e5i)11-s + (−8.95e5 + 3.70e5i)12-s − 7.21e4i·13-s + (−2.13e5 − 5.14e5i)14-s + (−1.03e6 − 1.03e6i)15-s − 3.88e5·16-s + (55.2 − 5.85e6i)17-s + ⋯ |
L(s) = 1 | + (0.413 + 0.413i)2-s + (−0.652 − 1.57i)3-s − 0.658i·4-s + (0.270 − 0.112i)5-s + (0.381 − 0.920i)6-s + (−0.437 − 0.181i)7-s + (0.685 − 0.685i)8-s + (−1.35 + 1.35i)9-s + (0.158 + 0.0655i)10-s + (−0.398 + 0.961i)11-s + (−1.03 + 0.430i)12-s − 0.0539i·13-s + (−0.105 − 0.255i)14-s + (−0.353 − 0.353i)15-s − 0.0926·16-s + (9.43e−6 − 0.999i)17-s + ⋯ |
Λ(s)=(=(17s/2ΓC(s)L(s)(−0.997−0.0758i)Λ(12−s)
Λ(s)=(=(17s/2ΓC(s+11/2)L(s)(−0.997−0.0758i)Λ(1−s)
Degree: |
2 |
Conductor: |
17
|
Sign: |
−0.997−0.0758i
|
Analytic conductor: |
13.0618 |
Root analytic conductor: |
3.61411 |
Motivic weight: |
11 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ17(15,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 17, ( :11/2), −0.997−0.0758i)
|
Particular Values
L(6) |
≈ |
0.0395133+1.03975i |
L(21) |
≈ |
0.0395133+1.03975i |
L(213) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 17 | 1+(−55.2+5.85e6i)T |
good | 2 | 1+(−18.6−18.6i)T+2.04e3iT2 |
| 3 | 1+(274.+663.i)T+(−1.25e5+1.25e5i)T2 |
| 5 | 1+(−1.89e3+783.i)T+(3.45e7−3.45e7i)T2 |
| 7 | 1+(1.94e4+8.05e3i)T+(1.39e9+1.39e9i)T2 |
| 11 | 1+(2.12e5−5.13e5i)T+(−2.01e11−2.01e11i)T2 |
| 13 | 1+7.21e4iT−1.79e12T2 |
| 19 | 1+(8.42e5+8.42e5i)T+1.16e14iT2 |
| 23 | 1+(6.36e6−1.53e7i)T+(−6.73e14−6.73e14i)T2 |
| 29 | 1+(−1.14e8+4.74e7i)T+(8.62e15−8.62e15i)T2 |
| 31 | 1+(6.49e7+1.56e8i)T+(−1.79e16+1.79e16i)T2 |
| 37 | 1+(−2.51e8−6.06e8i)T+(−1.25e17+1.25e17i)T2 |
| 41 | 1+(1.08e9+4.47e8i)T+(3.89e17+3.89e17i)T2 |
| 43 | 1+(−7.28e8+7.28e8i)T−9.29e17iT2 |
| 47 | 1+1.52e9iT−2.47e18T2 |
| 53 | 1+(6.14e7+6.14e7i)T+9.26e18iT2 |
| 59 | 1+(−5.34e9+5.34e9i)T−3.01e19iT2 |
| 61 | 1+(8.61e9+3.56e9i)T+(3.07e19+3.07e19i)T2 |
| 67 | 1−9.16e9T+1.22e20T2 |
| 71 | 1+(7.19e9+1.73e10i)T+(−1.63e20+1.63e20i)T2 |
| 73 | 1+(6.96e9−2.88e9i)T+(2.21e20−2.21e20i)T2 |
| 79 | 1+(−5.71e8+1.38e9i)T+(−5.28e20−5.28e20i)T2 |
| 83 | 1+(4.11e10+4.11e10i)T+1.28e21iT2 |
| 89 | 1−7.13e10iT−2.77e21T2 |
| 97 | 1+(1.07e11−4.46e10i)T+(5.05e21−5.05e21i)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
−15.46231840747114655590134218887, −13.78226013820324240423416312519, −13.08613090360479337036950690657, −11.71106245689108575070813402704, −9.937744144328234760183694810107, −7.46607450865417350851190921623, −6.46626061754489918142680643513, −5.18671651628577487271322447669, −1.89025526411099286678454561894, −0.40595011660416146554438920670,
3.07512005370217102347159550499, 4.34842658026822725533590912496, 5.89855448545637745890470616632, 8.603053285495085487771605128139, 10.24573782099057873898320153871, 11.21112805655233519260312807289, 12.60767823535101623440292056165, 14.25116929467510048186409152424, 15.88683361364560749025975972923, 16.57577734125648911426149811846