L(s) = 1 | + (0.866 − 0.5i)2-s + (2.44 + 1.41i)3-s + (−0.500 + 0.866i)4-s + (−0.448 − 2.19i)5-s + 2.82·6-s + 3i·8-s + (2.49 + 4.33i)9-s + (−1.48 − 1.67i)10-s + (−2.44 + 1.41i)12-s − 4.24i·13-s + (2 − 5.99i)15-s + (0.500 + 0.866i)16-s + (−3.67 − 2.12i)17-s + (4.33 + 2.5i)18-s + (1.41 + 2.44i)19-s + (2.12 + 0.707i)20-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (1.41 + 0.816i)3-s + (−0.250 + 0.433i)4-s + (−0.200 − 0.979i)5-s + 1.15·6-s + 1.06i·8-s + (0.833 + 1.44i)9-s + (−0.469 − 0.529i)10-s + (−0.707 + 0.408i)12-s − 1.17i·13-s + (0.516 − 1.54i)15-s + (0.125 + 0.216i)16-s + (−0.891 − 0.514i)17-s + (1.02 + 0.589i)18-s + (0.324 + 0.561i)19-s + (0.474 + 0.158i)20-s + ⋯ |
Λ(s)=(=(245s/2ΓC(s)L(s)(0.962−0.271i)Λ(2−s)
Λ(s)=(=(245s/2ΓC(s+1/2)L(s)(0.962−0.271i)Λ(1−s)
Degree: |
2 |
Conductor: |
245
= 5⋅72
|
Sign: |
0.962−0.271i
|
Analytic conductor: |
1.95633 |
Root analytic conductor: |
1.39869 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ245(214,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 245, ( :1/2), 0.962−0.271i)
|
Particular Values
L(1) |
≈ |
2.20239+0.304163i |
L(21) |
≈ |
2.20239+0.304163i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(0.448+2.19i)T |
| 7 | 1 |
good | 2 | 1+(−0.866+0.5i)T+(1−1.73i)T2 |
| 3 | 1+(−2.44−1.41i)T+(1.5+2.59i)T2 |
| 11 | 1+(−5.5−9.52i)T2 |
| 13 | 1+4.24iT−13T2 |
| 17 | 1+(3.67+2.12i)T+(8.5+14.7i)T2 |
| 19 | 1+(−1.41−2.44i)T+(−9.5+16.4i)T2 |
| 23 | 1+(3.46−2i)T+(11.5−19.9i)T2 |
| 29 | 1+29T2 |
| 31 | 1+(−2.82+4.89i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−5.19+3i)T+(18.5−32.0i)T2 |
| 41 | 1+4.24T+41T2 |
| 43 | 1−43T2 |
| 47 | 1+(23.5−40.7i)T2 |
| 53 | 1+(6.92+4i)T+(26.5+45.8i)T2 |
| 59 | 1+(4.24−7.34i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−4.94−8.57i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−10.3−6i)T+(33.5+58.0i)T2 |
| 71 | 1−12T+71T2 |
| 73 | 1+(11.0+6.36i)T+(36.5+63.2i)T2 |
| 79 | 1+(−6−10.3i)T+(−39.5+68.4i)T2 |
| 83 | 1−8.48iT−83T2 |
| 89 | 1+(−2.12−3.67i)T+(−44.5+77.0i)T2 |
| 97 | 1+4.24iT−97T2 |
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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
−12.37977353988219624470065449624, −11.33265278059940207005371497732, −9.969479765436604935495522424797, −9.169864961539175430110744365320, −8.259211887974300560475467237654, −7.79854165026498520128286149769, −5.44412810668871832723278910550, −4.40802441655419873847173313642, −3.63532866849199310705853611054, −2.46762104220079143664097961321,
1.99296762216752513172282824167, 3.35755073767409733152042992505, 4.48810055562412327469329330369, 6.43064286418123012046063774365, 6.82698723276605308888962268806, 7.994189159490589946696554177023, 9.042500187070389005217891172872, 9.922935020890104483901921803551, 11.17101314588107632546454601131, 12.41648181675204245904693330597