L(s) = 1 | + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (4.80 − 1.38i)5-s + (11.2 − 3.02i)7-s + (−1.99 + 2i)8-s + (−6.05 + 3.65i)10-s + (1.29 − 2.24i)11-s + (−18.3 − 4.90i)13-s + (−14.3 + 8.26i)14-s + (1.99 − 3.46i)16-s + (6.98 + 6.98i)17-s − 15.8i·19-s + (6.93 − 7.20i)20-s + (−0.950 + 3.54i)22-s + (2.99 + 0.802i)23-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.433 − 0.250i)4-s + (0.960 − 0.277i)5-s + (1.61 − 0.432i)7-s + (−0.249 + 0.250i)8-s + (−0.605 + 0.365i)10-s + (0.118 − 0.204i)11-s + (−1.40 − 0.377i)13-s + (−1.02 + 0.590i)14-s + (0.124 − 0.216i)16-s + (0.410 + 0.410i)17-s − 0.833i·19-s + (0.346 − 0.360i)20-s + (−0.0431 + 0.161i)22-s + (0.130 + 0.0349i)23-s + ⋯ |
Λ(s)=(=(270s/2ΓC(s)L(s)(0.912+0.408i)Λ(3−s)
Λ(s)=(=(270s/2ΓC(s+1)L(s)(0.912+0.408i)Λ(1−s)
Degree: |
2 |
Conductor: |
270
= 2⋅33⋅5
|
Sign: |
0.912+0.408i
|
Analytic conductor: |
7.35696 |
Root analytic conductor: |
2.71237 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ270(127,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 270, ( :1), 0.912+0.408i)
|
Particular Values
L(23) |
≈ |
1.52678−0.325960i |
L(21) |
≈ |
1.52678−0.325960i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.36−0.366i)T |
| 3 | 1 |
| 5 | 1+(−4.80+1.38i)T |
good | 7 | 1+(−11.2+3.02i)T+(42.4−24.5i)T2 |
| 11 | 1+(−1.29+2.24i)T+(−60.5−104.i)T2 |
| 13 | 1+(18.3+4.90i)T+(146.+84.5i)T2 |
| 17 | 1+(−6.98−6.98i)T+289iT2 |
| 19 | 1+15.8iT−361T2 |
| 23 | 1+(−2.99−0.802i)T+(458.+264.5i)T2 |
| 29 | 1+(−27.9−16.1i)T+(420.5+728.i)T2 |
| 31 | 1+(8.35+14.4i)T+(−480.5+832.i)T2 |
| 37 | 1+(−20.7−20.7i)T+1.36e3iT2 |
| 41 | 1+(30.6+53.0i)T+(−840.5+1.45e3i)T2 |
| 43 | 1+(−16.7−62.4i)T+(−1.60e3+924.5i)T2 |
| 47 | 1+(−4.44+1.19i)T+(1.91e3−1.10e3i)T2 |
| 53 | 1+(−19.2+19.2i)T−2.80e3iT2 |
| 59 | 1+(2.53−1.46i)T+(1.74e3−3.01e3i)T2 |
| 61 | 1+(−45.1+78.1i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(31.9−119.i)T+(−3.88e3−2.24e3i)T2 |
| 71 | 1+96.4T+5.04e3T2 |
| 73 | 1+(−2.64+2.64i)T−5.32e3iT2 |
| 79 | 1+(−20.6−11.9i)T+(3.12e3+5.40e3i)T2 |
| 83 | 1+(−28.1−104.i)T+(−5.96e3+3.44e3i)T2 |
| 89 | 1+51.4iT−7.92e3T2 |
| 97 | 1+(124.−33.4i)T+(8.14e3−4.70e3i)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
−11.42434159845802994528801733685, −10.53274802853099032059707179030, −9.763971751097286697944083422302, −8.699083426867103294216964572719, −7.86291792575903577192041247223, −6.88032885196617607115379452180, −5.47410111724003400660868321660, −4.69264259461436477816903411957, −2.44367878276576065441566105331, −1.14994256659498109504369896035,
1.59052438784475340110010287434, 2.57489751960674457437057035364, 4.67511462419559167348767770550, 5.67618644498229001348386693054, 7.05402114130424852166885369121, 7.952146210904444248693727559199, 8.984711834630887404959757048252, 9.885682087361845379312426227819, 10.65123495142702996377534484317, 11.75673713791915381410814806144