L(s) = 1 | + (−0.849 − 1.50i)3-s − 1.58·5-s + (−1.80 + 1.93i)7-s + (−1.55 + 2.56i)9-s + 5.17·11-s + (−0.681 + 1.18i)13-s + (1.34 + 2.38i)15-s + (−2.30 + 3.99i)17-s + (0.0321 + 0.0557i)19-s + (4.45 + 1.08i)21-s + 6.74·23-s − 2.49·25-s + (5.19 + 0.166i)27-s + (4.70 + 8.15i)29-s + (1.33 + 2.30i)31-s + ⋯ |
L(s) = 1 | + (−0.490 − 0.871i)3-s − 0.707·5-s + (−0.683 + 0.729i)7-s + (−0.518 + 0.855i)9-s + 1.55·11-s + (−0.189 + 0.327i)13-s + (0.347 + 0.616i)15-s + (−0.559 + 0.969i)17-s + (0.00738 + 0.0127i)19-s + (0.971 + 0.237i)21-s + 1.40·23-s − 0.499·25-s + (0.999 + 0.0320i)27-s + (0.874 + 1.51i)29-s + (0.239 + 0.414i)31-s + ⋯ |
Λ(s)=(=(504s/2ΓC(s)L(s)(0.615−0.788i)Λ(2−s)
Λ(s)=(=(504s/2ΓC(s+1/2)L(s)(0.615−0.788i)Λ(1−s)
Degree: |
2 |
Conductor: |
504
= 23⋅32⋅7
|
Sign: |
0.615−0.788i
|
Analytic conductor: |
4.02446 |
Root analytic conductor: |
2.00610 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ504(193,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 504, ( :1/2), 0.615−0.788i)
|
Particular Values
L(1) |
≈ |
0.733959+0.357997i |
L(21) |
≈ |
0.733959+0.357997i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(0.849+1.50i)T |
| 7 | 1+(1.80−1.93i)T |
good | 5 | 1+1.58T+5T2 |
| 11 | 1−5.17T+11T2 |
| 13 | 1+(0.681−1.18i)T+(−6.5−11.2i)T2 |
| 17 | 1+(2.30−3.99i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−0.0321−0.0557i)T+(−9.5+16.4i)T2 |
| 23 | 1−6.74T+23T2 |
| 29 | 1+(−4.70−8.15i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−1.33−2.30i)T+(−15.5+26.8i)T2 |
| 37 | 1+(−0.880−1.52i)T+(−18.5+32.0i)T2 |
| 41 | 1+(0.858−1.48i)T+(−20.5−35.5i)T2 |
| 43 | 1+(5.12+8.86i)T+(−21.5+37.2i)T2 |
| 47 | 1+(2.60−4.51i)T+(−23.5−40.7i)T2 |
| 53 | 1+(0.479−0.831i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−4.66−8.08i)T+(−29.5+51.0i)T2 |
| 61 | 1+(7.19−12.4i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−6.24−10.8i)T+(−33.5+58.0i)T2 |
| 71 | 1+4.49T+71T2 |
| 73 | 1+(0.941−1.63i)T+(−36.5−63.2i)T2 |
| 79 | 1+(3.26−5.65i)T+(−39.5−68.4i)T2 |
| 83 | 1+(5.08+8.81i)T+(−41.5+71.8i)T2 |
| 89 | 1+(4.12+7.14i)T+(−44.5+77.0i)T2 |
| 97 | 1+(7.26+12.5i)T+(−48.5+84.0i)T2 |
show more | |
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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
−11.38192137025946072762885685337, −10.29943398375088834731277474253, −8.944130019025096627744392079529, −8.531090515041616292902378681605, −7.06663776422068291615322971520, −6.66718284139504054001632182899, −5.64861427986854352048461803932, −4.33551520391233702449658011331, −3.04062394653424015772146430382, −1.44892528034718364303863730861,
0.56512606423638210165611545500, 3.16942768427714371386764826414, 4.05707480017478402401439059570, 4.83469434201374006061754449397, 6.31243749467952376073227507176, 6.90549968483241425623329616303, 8.156095512811796221045969253780, 9.409293127231843209476268289330, 9.707333867463478204937192525423, 10.94038785014325519540222379330