L(s) = 1 | + (−1.28 − 2.23i)2-s + (−0.5 + 0.866i)3-s + (−2.32 + 4.02i)4-s + (−1.82 − 3.15i)5-s + 2.57·6-s + (0.967 − 2.46i)7-s + 6.80·8-s + (−0.499 − 0.866i)9-s + (−4.69 + 8.12i)10-s + (2.06 − 3.58i)11-s + (−2.32 − 4.02i)12-s − 4.13·13-s + (−6.74 + 1.01i)14-s + 3.64·15-s + (−4.13 − 7.15i)16-s + (−1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.911 − 1.57i)2-s + (−0.288 + 0.499i)3-s + (−1.16 + 2.01i)4-s + (−0.814 − 1.41i)5-s + 1.05·6-s + (0.365 − 0.930i)7-s + 2.40·8-s + (−0.166 − 0.288i)9-s + (−1.48 + 2.57i)10-s + (0.623 − 1.08i)11-s + (−0.670 − 1.16i)12-s − 1.14·13-s + (−1.80 + 0.270i)14-s + 0.940·15-s + (−1.03 − 1.78i)16-s + (−0.363 + 0.630i)17-s + ⋯ |
Λ(s)=(=(399s/2ΓC(s)L(s)(−0.0857−0.996i)Λ(2−s)
Λ(s)=(=(399s/2ΓC(s+1/2)L(s)(−0.0857−0.996i)Λ(1−s)
Degree: |
2 |
Conductor: |
399
= 3⋅7⋅19
|
Sign: |
−0.0857−0.996i
|
Analytic conductor: |
3.18603 |
Root analytic conductor: |
1.78494 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ399(172,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 399, ( :1/2), −0.0857−0.996i)
|
Particular Values
L(1) |
≈ |
0.172963+0.188491i |
L(21) |
≈ |
0.172963+0.188491i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.5−0.866i)T |
| 7 | 1+(−0.967+2.46i)T |
| 19 | 1+(−0.5−0.866i)T |
good | 2 | 1+(1.28+2.23i)T+(−1+1.73i)T2 |
| 5 | 1+(1.82+3.15i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−2.06+3.58i)T+(−5.5−9.52i)T2 |
| 13 | 1+4.13T+13T2 |
| 17 | 1+(1.5−2.59i)T+(−8.5−14.7i)T2 |
| 23 | 1+(−1.41−2.45i)T+(−11.5+19.9i)T2 |
| 29 | 1+8.66T+29T2 |
| 31 | 1+(4.14−7.18i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−0.172−0.298i)T+(−18.5+32.0i)T2 |
| 41 | 1−3.31T+41T2 |
| 43 | 1−6.00T+43T2 |
| 47 | 1+(4.35+7.54i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−2.50+4.34i)T+(−26.5−45.8i)T2 |
| 59 | 1+(3.00−5.20i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−0.181−0.314i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−4.24+7.35i)T+(−33.5−58.0i)T2 |
| 71 | 1+5.88T+71T2 |
| 73 | 1+(−5.50+9.53i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−5.39−9.33i)T+(−39.5+68.4i)T2 |
| 83 | 1+17.3T+83T2 |
| 89 | 1+(−5.32−9.22i)T+(−44.5+77.0i)T2 |
| 97 | 1−1.21T+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
−10.83456397989575727558500373540, −9.712609092269909008395188070745, −9.005030736297470786050633881496, −8.301361180689588999399025096679, −7.38666349674295633350250507246, −5.21730333383282753861490840072, −4.15347758796463648473387294337, −3.55006981708457487248284724051, −1.46495551932363390506282059762, −0.24717407924063765583436559115,
2.34797165912495807673112095096, 4.52037087101876579045664366386, 5.70165795762043367771054835620, 6.68318842251652829665337800431, 7.39039340241357597664387242651, 7.72159554481953050398378171148, 9.095682162857757446499945788915, 9.746792465279353292941149241943, 10.99294561282799894868369121046, 11.70004107320202285065648597597