L(s) = 1 | + (7.51 − 2.73i)2-s + (37.5 + 27.9i)3-s + (49.0 − 41.1i)4-s + (−21.4 − 121. i)5-s + (358. + 107. i)6-s + (674. + 565. i)7-s + (256. − 443. i)8-s + (629. + 2.09e3i)9-s + (−494. − 856. i)10-s + (854. − 4.84e3i)11-s + (2.98e3 − 175. i)12-s + (8.39e3 + 3.05e3i)13-s + (6.61e3 + 2.40e3i)14-s + (2.59e3 − 5.16e3i)15-s + (711. − 4.03e3i)16-s + (2.81e3 + 4.88e3i)17-s + ⋯ |
L(s) = 1 | + (0.664 − 0.241i)2-s + (0.802 + 0.596i)3-s + (0.383 − 0.321i)4-s + (−0.0767 − 0.435i)5-s + (0.677 + 0.202i)6-s + (0.743 + 0.623i)7-s + (0.176 − 0.306i)8-s + (0.287 + 0.957i)9-s + (−0.156 − 0.270i)10-s + (0.193 − 1.09i)11-s + (0.499 − 0.0293i)12-s + (1.06 + 0.385i)13-s + (0.644 + 0.234i)14-s + (0.198 − 0.395i)15-s + (0.0434 − 0.246i)16-s + (0.139 + 0.241i)17-s + ⋯ |
Λ(s)=(=(54s/2ΓC(s)L(s)(0.993−0.117i)Λ(8−s)
Λ(s)=(=(54s/2ΓC(s+7/2)L(s)(0.993−0.117i)Λ(1−s)
Degree: |
2 |
Conductor: |
54
= 2⋅33
|
Sign: |
0.993−0.117i
|
Analytic conductor: |
16.8687 |
Root analytic conductor: |
4.10716 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ54(7,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 54, ( :7/2), 0.993−0.117i)
|
Particular Values
L(4) |
≈ |
3.93728+0.232903i |
L(21) |
≈ |
3.93728+0.232903i |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−7.51+2.73i)T |
| 3 | 1+(−37.5−27.9i)T |
good | 5 | 1+(21.4+121.i)T+(−7.34e4+2.67e4i)T2 |
| 7 | 1+(−674.−565.i)T+(1.43e5+8.11e5i)T2 |
| 11 | 1+(−854.+4.84e3i)T+(−1.83e7−6.66e6i)T2 |
| 13 | 1+(−8.39e3−3.05e3i)T+(4.80e7+4.03e7i)T2 |
| 17 | 1+(−2.81e3−4.88e3i)T+(−2.05e8+3.55e8i)T2 |
| 19 | 1+(1.59e4−2.75e4i)T+(−4.46e8−7.74e8i)T2 |
| 23 | 1+(−1.27e4+1.07e4i)T+(5.91e8−3.35e9i)T2 |
| 29 | 1+(1.97e4−7.18e3i)T+(1.32e10−1.10e10i)T2 |
| 31 | 1+(−1.06e5+8.95e4i)T+(4.77e9−2.70e10i)T2 |
| 37 | 1+(1.79e5+3.11e5i)T+(−4.74e10+8.22e10i)T2 |
| 41 | 1+(3.56e5+1.29e5i)T+(1.49e11+1.25e11i)T2 |
| 43 | 1+(1.33e5−7.56e5i)T+(−2.55e11−9.29e10i)T2 |
| 47 | 1+(1.61e5+1.35e5i)T+(8.79e10+4.98e11i)T2 |
| 53 | 1+1.41e6T+1.17e12T2 |
| 59 | 1+(−4.82e4−2.73e5i)T+(−2.33e12+8.51e11i)T2 |
| 61 | 1+(1.20e6+1.01e6i)T+(5.45e11+3.09e12i)T2 |
| 67 | 1+(3.54e6+1.29e6i)T+(4.64e12+3.89e12i)T2 |
| 71 | 1+(1.63e6+2.82e6i)T+(−4.54e12+7.87e12i)T2 |
| 73 | 1+(2.60e6−4.51e6i)T+(−5.52e12−9.56e12i)T2 |
| 79 | 1+(5.50e6−2.00e6i)T+(1.47e13−1.23e13i)T2 |
| 83 | 1+(−4.50e6+1.64e6i)T+(2.07e13−1.74e13i)T2 |
| 89 | 1+(−5.12e6+8.88e6i)T+(−2.21e13−3.83e13i)T2 |
| 97 | 1+(5.91e5−3.35e6i)T+(−7.59e13−2.76e13i)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
−14.02635060900110344812484727389, −12.94154943154023157097090064311, −11.56380045048167224031169004158, −10.56733625550179027774590093817, −8.929459537471588635278844038416, −8.180129619285511830652846938129, −5.98672553874172685492004478485, −4.59799123930959605236190642355, −3.32864392701431517230078013432, −1.64590839086306980463551260776,
1.49696269255512845683171717630, 3.14695249326808788333524613224, 4.60325468028951629660523356637, 6.62367693031396903769911776749, 7.49288848137919931006270065106, 8.726928380780324291292485526201, 10.48720312109808068022794094436, 11.78419934644478579447047480558, 13.04196208905797946513048955134, 13.85228077528096144386789224676