L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.340 + 0.196i)3-s + (−0.499 + 0.866i)4-s + 1.77i·5-s + (−0.340 − 0.196i)6-s + (−2.18 − 1.26i)7-s − 0.999·8-s + (−1.42 + 2.46i)9-s + (−1.53 + 0.888i)10-s + (−2.98 + 1.72i)11-s − 0.393i·12-s + (2.99 − 2.01i)13-s − 2.52i·14-s + (−0.349 − 0.604i)15-s + (−0.5 − 0.866i)16-s + (−2.52 − 3.26i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.196 + 0.113i)3-s + (−0.249 + 0.433i)4-s + 0.794i·5-s + (−0.138 − 0.0802i)6-s + (−0.824 − 0.476i)7-s − 0.353·8-s + (−0.474 + 0.821i)9-s + (−0.486 + 0.280i)10-s + (−0.899 + 0.519i)11-s − 0.113i·12-s + (0.830 − 0.557i)13-s − 0.673i·14-s + (−0.0901 − 0.156i)15-s + (−0.125 − 0.216i)16-s + (−0.611 − 0.791i)17-s + ⋯ |
Λ(s)=(=(442s/2ΓC(s)L(s)(−0.998−0.0531i)Λ(2−s)
Λ(s)=(=(442s/2ΓC(s+1/2)L(s)(−0.998−0.0531i)Λ(1−s)
Degree: |
2 |
Conductor: |
442
= 2⋅13⋅17
|
Sign: |
−0.998−0.0531i
|
Analytic conductor: |
3.52938 |
Root analytic conductor: |
1.87866 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ442(237,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 442, ( :1/2), −0.998−0.0531i)
|
Particular Values
L(1) |
≈ |
0.0224242+0.843824i |
L(21) |
≈ |
0.0224242+0.843824i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5−0.866i)T |
| 13 | 1+(−2.99+2.01i)T |
| 17 | 1+(2.52+3.26i)T |
good | 3 | 1+(0.340−0.196i)T+(1.5−2.59i)T2 |
| 5 | 1−1.77iT−5T2 |
| 7 | 1+(2.18+1.26i)T+(3.5+6.06i)T2 |
| 11 | 1+(2.98−1.72i)T+(5.5−9.52i)T2 |
| 19 | 1+(3.66−6.34i)T+(−9.5−16.4i)T2 |
| 23 | 1+(0.147−0.0854i)T+(11.5−19.9i)T2 |
| 29 | 1+(3.57−2.06i)T+(14.5−25.1i)T2 |
| 31 | 1−9.12iT−31T2 |
| 37 | 1+(−9.10+5.25i)T+(18.5−32.0i)T2 |
| 41 | 1+(0.111−0.0644i)T+(20.5−35.5i)T2 |
| 43 | 1+(2.65−4.60i)T+(−21.5−37.2i)T2 |
| 47 | 1+2.05T+47T2 |
| 53 | 1+4.26T+53T2 |
| 59 | 1+(−0.0723+0.125i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−13.4−7.76i)T+(30.5+52.8i)T2 |
| 67 | 1+(−4.08−7.07i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−4.14−2.39i)T+(35.5+61.4i)T2 |
| 73 | 1−9.40iT−73T2 |
| 79 | 1+2.76iT−79T2 |
| 83 | 1−6.67T+83T2 |
| 89 | 1+(7.64+13.2i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−5.50−3.17i)T+(48.5+84.0i)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
−11.38810644098398156068042936950, −10.59673435627475298778336231181, −10.00970033102044973411923513490, −8.614888562860959067991730143197, −7.72231800405374542939796008634, −6.85784007480740348900440405014, −5.98350468552610859010900946814, −4.99916121078982036100515291464, −3.69376169257994322988373393043, −2.61355335306055761772646173358,
0.46032072159158067974973748637, 2.38303816423699569524615034014, 3.61825956219651770364218715273, 4.76071551368952434899738647675, 5.97274792302725122128051225728, 6.48418853584145105322217749038, 8.293985895550713431721496308105, 8.991276916429000272294978795197, 9.677846298890319283776370505052, 11.06115637751582926901872919101