L(s) = 1 | + (−0.373 − 0.646i)2-s + (−0.160 − 1.72i)3-s + (0.721 − 1.24i)4-s + (−1.18 + 2.04i)5-s + (−1.05 + 0.747i)6-s + (0.193 + 0.334i)7-s − 2.56·8-s + (−2.94 + 0.552i)9-s + 1.76·10-s + (−2.61 − 4.52i)11-s + (−2.27 − 1.04i)12-s + (0.822 − 1.42i)13-s + (0.144 − 0.249i)14-s + (3.71 + 1.70i)15-s + (−0.484 − 0.838i)16-s − 6.42·17-s + ⋯ |
L(s) = 1 | + (−0.263 − 0.456i)2-s + (−0.0925 − 0.995i)3-s + (0.360 − 0.624i)4-s + (−0.528 + 0.915i)5-s + (−0.430 + 0.304i)6-s + (0.0729 + 0.126i)7-s − 0.908·8-s + (−0.982 + 0.184i)9-s + 0.557·10-s + (−0.787 − 1.36i)11-s + (−0.655 − 0.301i)12-s + (0.228 − 0.395i)13-s + (0.0385 − 0.0667i)14-s + (0.960 + 0.441i)15-s + (−0.121 − 0.209i)16-s − 1.55·17-s + ⋯ |
Λ(s)=(=(369s/2ΓC(s)L(s)(−0.986−0.163i)Λ(2−s)
Λ(s)=(=(369s/2ΓC(s+1/2)L(s)(−0.986−0.163i)Λ(1−s)
Degree: |
2 |
Conductor: |
369
= 32⋅41
|
Sign: |
−0.986−0.163i
|
Analytic conductor: |
2.94647 |
Root analytic conductor: |
1.71653 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ369(247,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 369, ( :1/2), −0.986−0.163i)
|
Particular Values
L(1) |
≈ |
0.0540838+0.659058i |
L(21) |
≈ |
0.0540838+0.659058i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.160+1.72i)T |
| 41 | 1+(−0.5+0.866i)T |
good | 2 | 1+(0.373+0.646i)T+(−1+1.73i)T2 |
| 5 | 1+(1.18−2.04i)T+(−2.5−4.33i)T2 |
| 7 | 1+(−0.193−0.334i)T+(−3.5+6.06i)T2 |
| 11 | 1+(2.61+4.52i)T+(−5.5+9.52i)T2 |
| 13 | 1+(−0.822+1.42i)T+(−6.5−11.2i)T2 |
| 17 | 1+6.42T+17T2 |
| 19 | 1+3.07T+19T2 |
| 23 | 1+(−3.88+6.73i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−4.70−8.14i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−1.19+2.06i)T+(−15.5−26.8i)T2 |
| 37 | 1−4.99T+37T2 |
| 43 | 1+(2.80+4.85i)T+(−21.5+37.2i)T2 |
| 47 | 1+(2.83+4.90i)T+(−23.5+40.7i)T2 |
| 53 | 1−1.07T+53T2 |
| 59 | 1+(−3.07+5.33i)T+(−29.5−51.0i)T2 |
| 61 | 1+(4.78+8.28i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−2.78+4.83i)T+(−33.5−58.0i)T2 |
| 71 | 1−12.9T+71T2 |
| 73 | 1+12.2T+73T2 |
| 79 | 1+(−3.37−5.84i)T+(−39.5+68.4i)T2 |
| 83 | 1+(0.388+0.673i)T+(−41.5+71.8i)T2 |
| 89 | 1+7.01T+89T2 |
| 97 | 1+(−3.91−6.78i)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
−10.84747754002285675098546078649, −10.63864560701946791830408963404, −8.833672864662416876102416815141, −8.257427700417107211250279123761, −6.82990806954420064323069756481, −6.44293686925277708279316941385, −5.23699045660914557840235603401, −3.14330802278140980634825800207, −2.33909019527947211290884007322, −0.45783767292706726772607802352,
2.59184991744776144602601886357, 4.21984733787412630959778116484, 4.70243260144409285976424531455, 6.18209899875759285073434141654, 7.34908515355649000998761771378, 8.289229442211387680484301167573, 8.981410407430992548294664707938, 9.862524110258723558263663920400, 11.07443957396681617745128896013, 11.74126152968078163952954239492