L(s) = 1 | + (−0.984 − 0.715i)2-s + (−0.160 − 0.494i)4-s + (−0.240 + 0.174i)5-s + (1.32 + 4.08i)7-s + (−0.947 + 2.91i)8-s + 0.361·10-s + (−2.91 + 1.58i)11-s + (3.06 + 2.22i)13-s + (1.61 − 4.97i)14-s + (2.17 − 1.58i)16-s + (1.09 − 0.796i)17-s + (1.01 − 3.12i)19-s + (0.124 + 0.0907i)20-s + (4.00 + 0.516i)22-s + 4.75·23-s + ⋯ |
L(s) = 1 | + (−0.695 − 0.505i)2-s + (−0.0803 − 0.247i)4-s + (−0.107 + 0.0780i)5-s + (0.501 + 1.54i)7-s + (−0.334 + 1.03i)8-s + 0.114·10-s + (−0.877 + 0.479i)11-s + (0.849 + 0.616i)13-s + (0.431 − 1.32i)14-s + (0.544 − 0.395i)16-s + (0.265 − 0.193i)17-s + (0.233 − 0.717i)19-s + (0.0279 + 0.0202i)20-s + (0.853 + 0.110i)22-s + 0.991·23-s + ⋯ |
Λ(s)=(=(297s/2ΓC(s)L(s)(0.865−0.500i)Λ(2−s)
Λ(s)=(=(297s/2ΓC(s+1/2)L(s)(0.865−0.500i)Λ(1−s)
Degree: |
2 |
Conductor: |
297
= 33⋅11
|
Sign: |
0.865−0.500i
|
Analytic conductor: |
2.37155 |
Root analytic conductor: |
1.53998 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ297(136,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 297, ( :1/2), 0.865−0.500i)
|
Particular Values
L(1) |
≈ |
0.771489+0.206741i |
L(21) |
≈ |
0.771489+0.206741i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 11 | 1+(2.91−1.58i)T |
good | 2 | 1+(0.984+0.715i)T+(0.618+1.90i)T2 |
| 5 | 1+(0.240−0.174i)T+(1.54−4.75i)T2 |
| 7 | 1+(−1.32−4.08i)T+(−5.66+4.11i)T2 |
| 13 | 1+(−3.06−2.22i)T+(4.01+12.3i)T2 |
| 17 | 1+(−1.09+0.796i)T+(5.25−16.1i)T2 |
| 19 | 1+(−1.01+3.12i)T+(−15.3−11.1i)T2 |
| 23 | 1−4.75T+23T2 |
| 29 | 1+(−3.09−9.52i)T+(−23.4+17.0i)T2 |
| 31 | 1+(4.15+3.01i)T+(9.57+29.4i)T2 |
| 37 | 1+(−1.40−4.30i)T+(−29.9+21.7i)T2 |
| 41 | 1+(−1.87+5.78i)T+(−33.1−24.0i)T2 |
| 43 | 1+7.92T+43T2 |
| 47 | 1+(−1.47+4.55i)T+(−38.0−27.6i)T2 |
| 53 | 1+(−1.16−0.845i)T+(16.3+50.4i)T2 |
| 59 | 1+(−1.95−6.01i)T+(−47.7+34.6i)T2 |
| 61 | 1+(−9.41+6.84i)T+(18.8−58.0i)T2 |
| 67 | 1+2.68T+67T2 |
| 71 | 1+(5.17−3.75i)T+(21.9−67.5i)T2 |
| 73 | 1+(−0.511−1.57i)T+(−59.0+42.9i)T2 |
| 79 | 1+(5.52+4.01i)T+(24.4+75.1i)T2 |
| 83 | 1+(4.08−2.97i)T+(25.6−78.9i)T2 |
| 89 | 1+16.3T+89T2 |
| 97 | 1+(3.15+2.29i)T+(29.9+92.2i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.48682940789183208415484294556, −11.04421101684038642250251783772, −9.890853223101985435586295801236, −8.922909228783547162190756238761, −8.527592797265572361778288519719, −7.12332697045744564079837166136, −5.61154290143614638431197659922, −5.00718406927905269395364563544, −2.90606852468253521906509412838, −1.70612263911683837954246862731,
0.801072296714126626768235543596, 3.34631089723658753843937905446, 4.37983592476870187644699415692, 5.93085959464424030049380085543, 7.15297536364385771094054690243, 7.980911085781261067837332210916, 8.408095727116535602869975035515, 9.884631443706908644929403654539, 10.52110286247959738377822281883, 11.46887939393595980341423029245