L(s) = 1 | + (1.93 + 1.62i)2-s + (−1.11 − 1.32i)3-s + (0.766 + 4.34i)4-s − 4.38i·6-s + (−0.532 + 3.01i)7-s + (−3.05 + 5.28i)8-s + (−0.520 + 2.95i)9-s + (−5.29 + 1.92i)11-s + (4.91 − 5.85i)12-s + (3.23 − 2.71i)13-s + (−5.94 + 4.98i)14-s + (−6.23 + 2.27i)16-s + (0.826 + 1.43i)17-s + (−5.81 + 4.88i)18-s + (−0.120 + 0.208i)19-s + ⋯ |
L(s) = 1 | + (1.37 + 1.15i)2-s + (−0.642 − 0.766i)3-s + (0.383 + 2.17i)4-s − 1.79i·6-s + (−0.201 + 1.14i)7-s + (−1.07 + 1.86i)8-s + (−0.173 + 0.984i)9-s + (−1.59 + 0.581i)11-s + (1.41 − 1.68i)12-s + (0.898 − 0.753i)13-s + (−1.58 + 1.33i)14-s + (−1.55 + 0.567i)16-s + (0.200 + 0.347i)17-s + (−1.37 + 1.15i)18-s + (−0.0276 + 0.0479i)19-s + ⋯ |
Λ(s)=(=(675s/2ΓC(s)L(s)(−0.835−0.549i)Λ(2−s)
Λ(s)=(=(675s/2ΓC(s+1/2)L(s)(−0.835−0.549i)Λ(1−s)
Degree: |
2 |
Conductor: |
675
= 33⋅52
|
Sign: |
−0.835−0.549i
|
Analytic conductor: |
5.38990 |
Root analytic conductor: |
2.32161 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(151,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 675, ( :1/2), −0.835−0.549i)
|
Particular Values
L(1) |
≈ |
0.597980+1.99739i |
L(21) |
≈ |
0.597980+1.99739i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.11+1.32i)T |
| 5 | 1 |
good | 2 | 1+(−1.93−1.62i)T+(0.347+1.96i)T2 |
| 7 | 1+(0.532−3.01i)T+(−6.57−2.39i)T2 |
| 11 | 1+(5.29−1.92i)T+(8.42−7.07i)T2 |
| 13 | 1+(−3.23+2.71i)T+(2.25−12.8i)T2 |
| 17 | 1+(−0.826−1.43i)T+(−8.5+14.7i)T2 |
| 19 | 1+(0.120−0.208i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−1.29−7.34i)T+(−21.6+7.86i)T2 |
| 29 | 1+(5.90+4.95i)T+(5.03+28.5i)T2 |
| 31 | 1+(−0.858−4.86i)T+(−29.1+10.6i)T2 |
| 37 | 1+(−1.24−2.15i)T+(−18.5+32.0i)T2 |
| 41 | 1+(0.109−0.0918i)T+(7.11−40.3i)T2 |
| 43 | 1+(−0.705+0.256i)T+(32.9−27.6i)T2 |
| 47 | 1+(−0.807+4.58i)T+(−44.1−16.0i)T2 |
| 53 | 1−12.1T+53T2 |
| 59 | 1+(−4.45−1.62i)T+(45.1+37.9i)T2 |
| 61 | 1+(−2.41+13.6i)T+(−57.3−20.8i)T2 |
| 67 | 1+(−5.64+4.73i)T+(11.6−65.9i)T2 |
| 71 | 1+(−2.45−4.24i)T+(−35.5+61.4i)T2 |
| 73 | 1+(−0.113+0.196i)T+(−36.5−63.2i)T2 |
| 79 | 1+(7.53+6.32i)T+(13.7+77.7i)T2 |
| 83 | 1+(6.78+5.69i)T+(14.4+81.7i)T2 |
| 89 | 1+(3.33−5.76i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−8.95+3.26i)T+(74.3−62.3i)T2 |
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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
−11.27807089733539838922890670983, −10.13199253597808307650265583725, −8.554105394763295256181889888185, −7.83619642395598822256986537163, −7.18638468179263059777562326953, −6.04473596252506159897046195666, −5.56367814293490797001744533372, −5.02041862030733594152870550675, −3.45829288065826710324307098404, −2.30829941676216497058240199911,
0.76317113316738554600419595630, 2.65024887684641983447561909864, 3.73206515307309049690443858974, 4.35337154258135707798419040943, 5.30966986742205641358240483983, 6.04473633705306742284023103118, 7.14232389227362843987232459369, 8.715858328789783011095538775293, 9.951924989627238695457421963068, 10.52666416635542442170747550575