L(s) = 1 | + (1 + i)2-s + 2i·4-s + (4.07 + 2.89i)5-s + (1.68 + 1.68i)7-s + (−2 + 2i)8-s + (1.18 + 6.97i)10-s + 6.57·11-s + (−3.12 + 3.12i)13-s + 3.37i·14-s − 4·16-s + (−4.54 − 4.54i)17-s + 27.6i·19-s + (−5.79 + 8.15i)20-s + (6.57 + 6.57i)22-s + (13.7 − 13.7i)23-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.815 + 0.579i)5-s + (0.241 + 0.241i)7-s + (−0.250 + 0.250i)8-s + (0.118 + 0.697i)10-s + 0.597·11-s + (−0.240 + 0.240i)13-s + 0.241i·14-s − 0.250·16-s + (−0.267 − 0.267i)17-s + 1.45i·19-s + (−0.289 + 0.407i)20-s + (0.298 + 0.298i)22-s + (0.598 − 0.598i)23-s + ⋯ |
Λ(s)=(=(270s/2ΓC(s)L(s)(0.0640−0.997i)Λ(3−s)
Λ(s)=(=(270s/2ΓC(s+1)L(s)(0.0640−0.997i)Λ(1−s)
Degree: |
2 |
Conductor: |
270
= 2⋅33⋅5
|
Sign: |
0.0640−0.997i
|
Analytic conductor: |
7.35696 |
Root analytic conductor: |
2.71237 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ270(217,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 270, ( :1), 0.0640−0.997i)
|
Particular Values
L(23) |
≈ |
1.73149+1.62398i |
L(21) |
≈ |
1.73149+1.62398i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1−i)T |
| 3 | 1 |
| 5 | 1+(−4.07−2.89i)T |
good | 7 | 1+(−1.68−1.68i)T+49iT2 |
| 11 | 1−6.57T+121T2 |
| 13 | 1+(3.12−3.12i)T−169iT2 |
| 17 | 1+(4.54+4.54i)T+289iT2 |
| 19 | 1−27.6iT−361T2 |
| 23 | 1+(−13.7+13.7i)T−529iT2 |
| 29 | 1+10.9iT−841T2 |
| 31 | 1+36.4T+961T2 |
| 37 | 1+(−43.7−43.7i)T+1.36e3iT2 |
| 41 | 1−44.3T+1.68e3T2 |
| 43 | 1+(−2.47+2.47i)T−1.84e3iT2 |
| 47 | 1+(62.4+62.4i)T+2.20e3iT2 |
| 53 | 1+(−29.0+29.0i)T−2.80e3iT2 |
| 59 | 1+41.6iT−3.48e3T2 |
| 61 | 1+42.5T+3.72e3T2 |
| 67 | 1+(67.5+67.5i)T+4.48e3iT2 |
| 71 | 1−47.6T+5.04e3T2 |
| 73 | 1+(−74.6+74.6i)T−5.32e3iT2 |
| 79 | 1+78.3iT−6.24e3T2 |
| 83 | 1+(−60.7+60.7i)T−6.88e3iT2 |
| 89 | 1−165.iT−7.92e3T2 |
| 97 | 1+(86.6+86.6i)T+9.40e3iT2 |
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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
−12.00794468341653283773841403926, −11.08888873901943947549532048990, −9.963265041699846009162459678723, −9.058592898015461302344271532801, −7.87980946331897309372401293452, −6.74605802370465633351659993238, −5.99893421934297209318266331829, −4.88492494782244520693807453934, −3.48038736584617824823912352944, −2.01690984077805322143531039911,
1.15590725822500198273414872454, 2.59915183235057420001729411581, 4.20761931143158167274027438257, 5.18633436167263272505344243061, 6.21983241766879191200862818854, 7.44202603578085256840308415098, 9.007541719730084408540429519737, 9.467329767062754079756331186491, 10.74905171824715883606056805481, 11.39213193302233657209898666090