L(s) = 1 | + (−0.867 + 1.11i)2-s + (−0.496 − 1.93i)4-s + (−2.93 + 1.69i)5-s + (−1.85 + 1.88i)7-s + (2.59 + 1.12i)8-s + (0.652 − 4.74i)10-s + (0.0932 + 0.0538i)11-s − 1.50i·13-s + (−0.504 − 3.70i)14-s + (−3.50 + 1.92i)16-s + (0.214 − 0.372i)17-s + (4.32 − 2.49i)19-s + (4.73 + 4.84i)20-s + (−0.141 + 0.0575i)22-s + (−4.56 − 7.90i)23-s + ⋯ |
L(s) = 1 | + (−0.613 + 0.789i)2-s + (−0.248 − 0.968i)4-s + (−1.31 + 0.757i)5-s + (−0.700 + 0.713i)7-s + (0.917 + 0.398i)8-s + (0.206 − 1.50i)10-s + (0.0281 + 0.0162i)11-s − 0.416i·13-s + (−0.134 − 0.990i)14-s + (−0.876 + 0.480i)16-s + (0.0520 − 0.0902i)17-s + (0.992 − 0.573i)19-s + (1.05 + 1.08i)20-s + (−0.0300 + 0.0122i)22-s + (−0.951 − 1.64i)23-s + ⋯ |
Λ(s)=(=(504s/2ΓC(s)L(s)(0.316+0.948i)Λ(2−s)
Λ(s)=(=(504s/2ΓC(s+1/2)L(s)(0.316+0.948i)Λ(1−s)
Degree: |
2 |
Conductor: |
504
= 23⋅32⋅7
|
Sign: |
0.316+0.948i
|
Analytic conductor: |
4.02446 |
Root analytic conductor: |
2.00610 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ504(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 504, ( :1/2), 0.316+0.948i)
|
Particular Values
L(1) |
≈ |
0.177759−0.128029i |
L(21) |
≈ |
0.177759−0.128029i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.867−1.11i)T |
| 3 | 1 |
| 7 | 1+(1.85−1.88i)T |
good | 5 | 1+(2.93−1.69i)T+(2.5−4.33i)T2 |
| 11 | 1+(−0.0932−0.0538i)T+(5.5+9.52i)T2 |
| 13 | 1+1.50iT−13T2 |
| 17 | 1+(−0.214+0.372i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−4.32+2.49i)T+(9.5−16.4i)T2 |
| 23 | 1+(4.56+7.90i)T+(−11.5+19.9i)T2 |
| 29 | 1+7.95iT−29T2 |
| 31 | 1+(0.393−0.680i)T+(−15.5−26.8i)T2 |
| 37 | 1+(7.68−4.43i)T+(18.5−32.0i)T2 |
| 41 | 1+8.59T+41T2 |
| 43 | 1+6.65iT−43T2 |
| 47 | 1+(−2.87−4.97i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−0.286−0.165i)T+(26.5+45.8i)T2 |
| 59 | 1+(−8.63−4.98i)T+(29.5+51.0i)T2 |
| 61 | 1+(−1.76+1.02i)T+(30.5−52.8i)T2 |
| 67 | 1+(2.79+1.61i)T+(33.5+58.0i)T2 |
| 71 | 1−8.72T+71T2 |
| 73 | 1+(4.38−7.59i)T+(−36.5−63.2i)T2 |
| 79 | 1+(0.785+1.36i)T+(−39.5+68.4i)T2 |
| 83 | 1+1.33iT−83T2 |
| 89 | 1+(3.62+6.27i)T+(−44.5+77.0i)T2 |
| 97 | 1+19.0T+97T2 |
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
−10.51566324492680142816034200784, −9.857918945755887084477007007532, −8.699818320080233207698942588278, −8.054305381440386660055374509985, −7.10174688672051445233686681092, −6.43063398577637997718309770159, −5.29550784973805049551844115047, −3.97476456861652622971427677980, −2.67208201001362048812581357466, −0.17214333770015165648542447230,
1.37339033117193819851098318986, 3.48084301078545562633091942831, 3.85604161523892774055619653539, 5.16963029138157883808859068845, 6.99632549850124128111491255106, 7.65030808024374377765706748716, 8.488465878876412805240505087492, 9.401050043079030261361676173480, 10.16175086900220401078079596281, 11.18716203618320213533854514602