L(s) = 1 | + (−0.866 + 0.5i)2-s + (1.21 − 1.23i)3-s + (0.499 − 0.866i)4-s + (2.73 + 1.57i)5-s + (−0.434 + 1.67i)6-s + (1.36 − 0.787i)7-s + 0.999i·8-s + (−0.0487 − 2.99i)9-s − 3.15·10-s + (−3.26 + 1.88i)11-s + (−0.461 − 1.66i)12-s + (−3.47 + 0.966i)13-s + (−0.787 + 1.36i)14-s + (5.27 − 1.45i)15-s + (−0.5 − 0.866i)16-s + 7.06·17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.701 − 0.712i)3-s + (0.249 − 0.433i)4-s + (1.22 + 0.706i)5-s + (−0.177 + 0.684i)6-s + (0.515 − 0.297i)7-s + 0.353i·8-s + (−0.0162 − 0.999i)9-s − 0.998·10-s + (−0.983 + 0.567i)11-s + (−0.133 − 0.481i)12-s + (−0.963 + 0.268i)13-s + (−0.210 + 0.364i)14-s + (1.36 − 0.376i)15-s + (−0.125 − 0.216i)16-s + 1.71·17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.998+0.0613i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.998+0.0613i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.998+0.0613i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), 0.998+0.0613i)
|
Particular Values
L(1) |
≈ |
1.36024−0.0417435i |
L(21) |
≈ |
1.36024−0.0417435i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.866−0.5i)T |
| 3 | 1+(−1.21+1.23i)T |
| 13 | 1+(3.47−0.966i)T |
good | 5 | 1+(−2.73−1.57i)T+(2.5+4.33i)T2 |
| 7 | 1+(−1.36+0.787i)T+(3.5−6.06i)T2 |
| 11 | 1+(3.26−1.88i)T+(5.5−9.52i)T2 |
| 17 | 1−7.06T+17T2 |
| 19 | 1+3.76iT−19T2 |
| 23 | 1+(1.84−3.20i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−0.109−0.189i)T+(−14.5+25.1i)T2 |
| 31 | 1+(2.65+1.53i)T+(15.5+26.8i)T2 |
| 37 | 1+0.292iT−37T2 |
| 41 | 1+(−6.39−3.69i)T+(20.5+35.5i)T2 |
| 43 | 1+(3.05+5.29i)T+(−21.5+37.2i)T2 |
| 47 | 1+(6.17−3.56i)T+(23.5−40.7i)T2 |
| 53 | 1+14.4T+53T2 |
| 59 | 1+(9.04+5.22i)T+(29.5+51.0i)T2 |
| 61 | 1+(−3.00−5.19i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−6.33−3.65i)T+(33.5+58.0i)T2 |
| 71 | 1+0.772iT−71T2 |
| 73 | 1+13.5iT−73T2 |
| 79 | 1+(−6.34−10.9i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−0.314+0.181i)T+(41.5−71.8i)T2 |
| 89 | 1−7.06iT−89T2 |
| 97 | 1+(−0.535+0.309i)T+(48.5−84.0i)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
−12.28900601578666752155789182507, −10.96518797330657632816068457512, −9.840878691215291256760115801013, −9.491150359066303318437397334111, −7.888627196093042073480478441155, −7.43657083650313929678038110742, −6.33671845116051119073235757062, −5.16532424263268325667529000304, −2.85889942830648546235385732899, −1.76409765835332206222012913033,
1.87602708748577297051869169750, 3.11372949108653322685213205989, 4.93881954450274170371056278386, 5.71739817690431645319953636788, 7.81554473651248109205300145001, 8.333273120087606100929493044489, 9.531785293306634158287182261048, 9.973303046699107096779236769654, 10.82515767816394440865436915401, 12.26278363757293837890398884475