L(s) = 1 | + (−0.797 + 2.30i)2-s + (0.458 + 0.888i)3-s + (−3.09 − 2.43i)4-s + (1.48 − 0.142i)5-s + (−2.41 + 0.346i)6-s + (2.64 + 0.114i)7-s + (3.98 − 2.56i)8-s + (−0.580 + 0.814i)9-s + (−0.859 + 3.54i)10-s + (5.25 − 1.81i)11-s + (0.746 − 3.87i)12-s + (1.14 + 3.89i)13-s + (−2.37 + 5.99i)14-s + (0.808 + 1.25i)15-s + (0.863 + 3.56i)16-s + (2.36 + 0.945i)17-s + ⋯ |
L(s) = 1 | + (−0.563 + 1.62i)2-s + (0.264 + 0.513i)3-s + (−1.54 − 1.21i)4-s + (0.665 − 0.0635i)5-s + (−0.985 + 0.141i)6-s + (0.999 + 0.0433i)7-s + (1.40 − 0.905i)8-s + (−0.193 + 0.271i)9-s + (−0.271 + 1.12i)10-s + (1.58 − 0.548i)11-s + (0.215 − 1.11i)12-s + (0.317 + 1.07i)13-s + (−0.633 + 1.60i)14-s + (0.208 + 0.324i)15-s + (0.215 + 0.890i)16-s + (0.572 + 0.229i)17-s + ⋯ |
Λ(s)=(=(483s/2ΓC(s)L(s)(−0.816−0.577i)Λ(2−s)
Λ(s)=(=(483s/2ΓC(s+1/2)L(s)(−0.816−0.577i)Λ(1−s)
Degree: |
2 |
Conductor: |
483
= 3⋅7⋅23
|
Sign: |
−0.816−0.577i
|
Analytic conductor: |
3.85677 |
Root analytic conductor: |
1.96386 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ483(10,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 483, ( :1/2), −0.816−0.577i)
|
Particular Values
L(1) |
≈ |
0.420805+1.32451i |
L(21) |
≈ |
0.420805+1.32451i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.458−0.888i)T |
| 7 | 1+(−2.64−0.114i)T |
| 23 | 1+(−0.501+4.76i)T |
good | 2 | 1+(0.797−2.30i)T+(−1.57−1.23i)T2 |
| 5 | 1+(−1.48+0.142i)T+(4.90−0.946i)T2 |
| 11 | 1+(−5.25+1.81i)T+(8.64−6.79i)T2 |
| 13 | 1+(−1.14−3.89i)T+(−10.9+7.02i)T2 |
| 17 | 1+(−2.36−0.945i)T+(12.3+11.7i)T2 |
| 19 | 1+(4.40−1.76i)T+(13.7−13.1i)T2 |
| 29 | 1+(0.100+0.701i)T+(−27.8+8.17i)T2 |
| 31 | 1+(−1.67+0.0797i)T+(30.8−2.94i)T2 |
| 37 | 1+(0.543+0.387i)T+(12.1+34.9i)T2 |
| 41 | 1+(5.32+2.43i)T+(26.8+30.9i)T2 |
| 43 | 1+(4.30−6.70i)T+(−17.8−39.1i)T2 |
| 47 | 1+(−1.90+1.10i)T+(23.5−40.7i)T2 |
| 53 | 1+(1.07+1.12i)T+(−2.52+52.9i)T2 |
| 59 | 1+(0.288+0.0700i)T+(52.4+27.0i)T2 |
| 61 | 1+(10.0+5.18i)T+(35.3+49.6i)T2 |
| 67 | 1+(2.58+13.4i)T+(−62.2+24.9i)T2 |
| 71 | 1+(2.97+3.43i)T+(−10.1+70.2i)T2 |
| 73 | 1+(−2.80+3.56i)T+(−17.2−70.9i)T2 |
| 79 | 1+(8.36−8.77i)T+(−3.75−78.9i)T2 |
| 83 | 1+(−2.45−5.38i)T+(−54.3+62.7i)T2 |
| 89 | 1+(0.806−16.9i)T+(−88.5−8.45i)T2 |
| 97 | 1+(−4.75+10.4i)T+(−63.5−73.3i)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
−11.11128789901034643375080733162, −10.02706452013603033948905400315, −9.143899229109399752833340186982, −8.662765555297787799606383787189, −7.88395420588078888708329432896, −6.55747973664589254608312156132, −6.10356577782973636933295095991, −4.91289349592141071006086108341, −4.00434244984890029039672725751, −1.63762037668329498201020355430,
1.24107773932101678758243742793, 1.98954480831316970178439700567, 3.29791437907042080185779227411, 4.42539589668325955606920905574, 5.86806457186866222648705506159, 7.23597539292603497943796268074, 8.344276629025128779080337117384, 8.995217199722636336395031376871, 9.888384064765295608831394102291, 10.59737858445064806139069262662