L(s) = 1 | + (4.38 − 2.52i)2-s + (−51.2 + 88.6i)4-s + (258. + 447. i)5-s + (−900. − 111. i)7-s + 1.16e3i·8-s + (2.26e3 + 1.30e3i)10-s + (−4.61e3 − 2.66e3i)11-s − 5.26e3i·13-s + (−4.22e3 + 1.78e3i)14-s + (−3.60e3 − 6.24e3i)16-s + (1.24e4 − 2.14e4i)17-s + (−1.31e4 + 7.58e3i)19-s − 5.28e4·20-s − 2.69e4·22-s + (−3.34e4 + 1.93e4i)23-s + ⋯ |
L(s) = 1 | + (0.387 − 0.223i)2-s + (−0.400 + 0.692i)4-s + (0.923 + 1.60i)5-s + (−0.992 − 0.122i)7-s + 0.804i·8-s + (0.715 + 0.413i)10-s + (−1.04 − 0.604i)11-s − 0.664i·13-s + (−0.411 + 0.174i)14-s + (−0.220 − 0.381i)16-s + (0.612 − 1.06i)17-s + (−0.439 + 0.253i)19-s − 1.47·20-s − 0.540·22-s + (−0.573 + 0.330i)23-s + ⋯ |
Λ(s)=(=(63s/2ΓC(s)L(s)(−0.996−0.0882i)Λ(8−s)
Λ(s)=(=(63s/2ΓC(s+7/2)L(s)(−0.996−0.0882i)Λ(1−s)
Degree: |
2 |
Conductor: |
63
= 32⋅7
|
Sign: |
−0.996−0.0882i
|
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), −0.996−0.0882i)
|
Particular Values
L(4) |
≈ |
0.0407364+0.921221i |
L(21) |
≈ |
0.0407364+0.921221i |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(900.+111.i)T |
good | 2 | 1+(−4.38+2.52i)T+(64−110.i)T2 |
| 5 | 1+(−258.−447.i)T+(−3.90e4+6.76e4i)T2 |
| 11 | 1+(4.61e3+2.66e3i)T+(9.74e6+1.68e7i)T2 |
| 13 | 1+5.26e3iT−6.27e7T2 |
| 17 | 1+(−1.24e4+2.14e4i)T+(−2.05e8−3.55e8i)T2 |
| 19 | 1+(1.31e4−7.58e3i)T+(4.46e8−7.74e8i)T2 |
| 23 | 1+(3.34e4−1.93e4i)T+(1.70e9−2.94e9i)T2 |
| 29 | 1−1.86e4iT−1.72e10T2 |
| 31 | 1+(−1.43e5−8.29e4i)T+(1.37e10+2.38e10i)T2 |
| 37 | 1+(−2.32e5−4.03e5i)T+(−4.74e10+8.22e10i)T2 |
| 41 | 1+4.66e5T+1.94e11T2 |
| 43 | 1+8.92e5T+2.71e11T2 |
| 47 | 1+(−3.95e4−6.84e4i)T+(−2.53e11+4.38e11i)T2 |
| 53 | 1+(1.81e5+1.04e5i)T+(5.87e11+1.01e12i)T2 |
| 59 | 1+(1.11e6−1.93e6i)T+(−1.24e12−2.15e12i)T2 |
| 61 | 1+(−2.06e6+1.19e6i)T+(1.57e12−2.72e12i)T2 |
| 67 | 1+(1.12e6−1.95e6i)T+(−3.03e12−5.24e12i)T2 |
| 71 | 1−1.19e5iT−9.09e12T2 |
| 73 | 1+(−8.45e4−4.88e4i)T+(5.52e12+9.56e12i)T2 |
| 79 | 1+(7.02e5+1.21e6i)T+(−9.60e12+1.66e13i)T2 |
| 83 | 1−1.04e6T+2.71e13T2 |
| 89 | 1+(7.51e5+1.30e6i)T+(−2.21e13+3.83e13i)T2 |
| 97 | 1−1.54e7iT−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
−13.62364799057132313523127814898, −13.35684939346327433210772123849, −11.86017203574326495767340869043, −10.51824204601392542675195861907, −9.771512117941808376691408170620, −8.013992310051067077284238882992, −6.71168820724203040840662061446, −5.45239534988942456752245231339, −3.26192391977792640937406494267, −2.73795764606513754916118380939,
0.27837172180552514955990269069, 1.84753978555022263172973002523, 4.32760514685581702686362761671, 5.41929673630405794981648448073, 6.34841484300804819471960030620, 8.450032134327731704270978982132, 9.608189584789226492628475380898, 10.16392755248985405798963039857, 12.43750597392742342827739224851, 13.02565959179878459444200362476