L(s) = 1 | + (−18.8 + 10.8i)2-s + (172. − 297. i)4-s + (38.2 + 66.1i)5-s + (884. + 201. i)7-s + 4.69e3i·8-s + (−1.43e3 − 830. i)10-s + (−338. − 195. i)11-s − 9.29e3i·13-s + (−1.88e4 + 5.82e3i)14-s + (−2.89e4 − 5.01e4i)16-s + (1.01e4 − 1.75e4i)17-s + (−4.12e4 + 2.38e4i)19-s + 2.62e4·20-s + 8.48e3·22-s + (−5.30e4 + 3.06e4i)23-s + ⋯ |
L(s) = 1 | + (−1.66 + 0.960i)2-s + (1.34 − 2.32i)4-s + (0.136 + 0.236i)5-s + (0.975 + 0.221i)7-s + 3.24i·8-s + (−0.454 − 0.262i)10-s + (−0.0766 − 0.0442i)11-s − 1.17i·13-s + (−1.83 + 0.567i)14-s + (−1.76 − 3.06i)16-s + (0.501 − 0.868i)17-s + (−1.37 + 0.796i)19-s + 0.734·20-s + 0.169·22-s + (−0.908 + 0.524i)23-s + ⋯ |
Λ(s)=(=(63s/2ΓC(s)L(s)(0.982+0.187i)Λ(8−s)
Λ(s)=(=(63s/2ΓC(s+7/2)L(s)(0.982+0.187i)Λ(1−s)
Degree: |
2 |
Conductor: |
63
= 32⋅7
|
Sign: |
0.982+0.187i
|
Analytic conductor: |
19.6802 |
Root analytic conductor: |
4.43624 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ63(17,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 63, ( :7/2), 0.982+0.187i)
|
Particular Values
L(4) |
≈ |
0.780956−0.0740053i |
L(21) |
≈ |
0.780956−0.0740053i |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(−884.−201.i)T |
good | 2 | 1+(18.8−10.8i)T+(64−110.i)T2 |
| 5 | 1+(−38.2−66.1i)T+(−3.90e4+6.76e4i)T2 |
| 11 | 1+(338.+195.i)T+(9.74e6+1.68e7i)T2 |
| 13 | 1+9.29e3iT−6.27e7T2 |
| 17 | 1+(−1.01e4+1.75e4i)T+(−2.05e8−3.55e8i)T2 |
| 19 | 1+(4.12e4−2.38e4i)T+(4.46e8−7.74e8i)T2 |
| 23 | 1+(5.30e4−3.06e4i)T+(1.70e9−2.94e9i)T2 |
| 29 | 1+1.36e5iT−1.72e10T2 |
| 31 | 1+(−2.25e5−1.30e5i)T+(1.37e10+2.38e10i)T2 |
| 37 | 1+(4.99e4+8.65e4i)T+(−4.74e10+8.22e10i)T2 |
| 41 | 1+5.63e4T+1.94e11T2 |
| 43 | 1−1.68e4T+2.71e11T2 |
| 47 | 1+(6.38e5+1.10e6i)T+(−2.53e11+4.38e11i)T2 |
| 53 | 1+(−8.22e5−4.74e5i)T+(5.87e11+1.01e12i)T2 |
| 59 | 1+(−6.91e5+1.19e6i)T+(−1.24e12−2.15e12i)T2 |
| 61 | 1+(−2.50e6+1.44e6i)T+(1.57e12−2.72e12i)T2 |
| 67 | 1+(−1.28e6+2.21e6i)T+(−3.03e12−5.24e12i)T2 |
| 71 | 1+4.17e6iT−9.09e12T2 |
| 73 | 1+(7.16e5+4.13e5i)T+(5.52e12+9.56e12i)T2 |
| 79 | 1+(−2.32e6−4.02e6i)T+(−9.60e12+1.66e13i)T2 |
| 83 | 1−6.11e6T+2.71e13T2 |
| 89 | 1+(1.09e6+1.90e6i)T+(−2.21e13+3.83e13i)T2 |
| 97 | 1+3.26e6iT−8.07e13T2 |
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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
−14.04418374632796369761416427115, −11.90330383202100227708819563629, −10.62807131962521145010290247271, −9.923396095348971651729065090184, −8.395537435699786318367622575628, −7.902832545589091251747881300777, −6.45388415086579962974953986896, −5.27834488388230950513279666913, −2.12298866478104845591365591478, −0.56559037413349577110976035761,
1.16863305820783200958322172542, 2.25734392161315708666424338642, 4.19991223373421822858691644502, 6.78714460448712760208400692358, 8.140431447549642904750027028544, 8.862838104758773746392277133336, 10.12797825306932529550471362059, 11.05403625976599338762787417425, 11.91944968819872576665433149626, 13.07230995752348330670605316839