L(s) = 1 | + (−0.227 − 0.394i)2-s + (−2.69 + 1.30i)3-s + (1.89 − 3.28i)4-s − 4.37i·5-s + (1.13 + 0.766i)6-s + (5.22 − 4.66i)7-s − 3.54·8-s + (5.57 − 7.06i)9-s + (−1.72 + 0.994i)10-s − 0.139·11-s + (−0.819 + 11.3i)12-s + (−1.71 + 0.987i)13-s + (−3.02 − 0.996i)14-s + (5.72 + 11.7i)15-s + (−6.77 − 11.7i)16-s + (−26.7 + 15.4i)17-s + ⋯ |
L(s) = 1 | + (−0.113 − 0.197i)2-s + (−0.899 + 0.436i)3-s + (0.474 − 0.821i)4-s − 0.874i·5-s + (0.188 + 0.127i)6-s + (0.745 − 0.666i)7-s − 0.443·8-s + (0.619 − 0.785i)9-s + (−0.172 + 0.0994i)10-s − 0.0126·11-s + (−0.0683 + 0.945i)12-s + (−0.131 + 0.0759i)13-s + (−0.216 − 0.0711i)14-s + (0.381 + 0.786i)15-s + (−0.423 − 0.733i)16-s + (−1.57 + 0.909i)17-s + ⋯ |
Λ(s)=(=(63s/2ΓC(s)L(s)(0.279+0.960i)Λ(3−s)
Λ(s)=(=(63s/2ΓC(s+1)L(s)(0.279+0.960i)Λ(1−s)
Degree: |
2 |
Conductor: |
63
= 32⋅7
|
Sign: |
0.279+0.960i
|
Analytic conductor: |
1.71662 |
Root analytic conductor: |
1.31020 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ63(61,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 63, ( :1), 0.279+0.960i)
|
Particular Values
L(23) |
≈ |
0.780005−0.585144i |
L(21) |
≈ |
0.780005−0.585144i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(2.69−1.30i)T |
| 7 | 1+(−5.22+4.66i)T |
good | 2 | 1+(0.227+0.394i)T+(−2+3.46i)T2 |
| 5 | 1+4.37iT−25T2 |
| 11 | 1+0.139T+121T2 |
| 13 | 1+(1.71−0.987i)T+(84.5−146.i)T2 |
| 17 | 1+(26.7−15.4i)T+(144.5−250.i)T2 |
| 19 | 1+(−25.2−14.5i)T+(180.5+312.i)T2 |
| 23 | 1−29.7T+529T2 |
| 29 | 1+(−7.28+12.6i)T+(−420.5−728.i)T2 |
| 31 | 1+(6.82+3.94i)T+(480.5+832.i)T2 |
| 37 | 1+(7.73−13.3i)T+(−684.5−1.18e3i)T2 |
| 41 | 1+(−0.747+0.431i)T+(840.5−1.45e3i)T2 |
| 43 | 1+(15.6−27.1i)T+(−924.5−1.60e3i)T2 |
| 47 | 1+(−58.0+33.4i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1+(−16.9−29.3i)T+(−1.40e3+2.43e3i)T2 |
| 59 | 1+(−57.4−33.1i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(35.9−20.7i)T+(1.86e3−3.22e3i)T2 |
| 67 | 1+(51.7−89.6i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1−86.4T+5.04e3T2 |
| 73 | 1+(28.6−16.5i)T+(2.66e3−4.61e3i)T2 |
| 79 | 1+(24.3+42.1i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(102.+59.4i)T+(3.44e3+5.96e3i)T2 |
| 89 | 1+(33.1+19.1i)T+(3.96e3+6.85e3i)T2 |
| 97 | 1+(−70.3−40.6i)T+(4.70e3+8.14e3i)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
−14.73071712215195852213788569244, −13.27876392844933561470213183255, −11.92095090628488893442976487367, −11.05404836568123462270829318187, −10.17346776170876205233491913977, −8.899025490177473410705230203711, −7.01359058065097700092259900178, −5.56168483101769852722907418885, −4.48118955746234351847596143558, −1.18183630518812640405228522140,
2.61957133700842662158381184626, 5.06383083171879071002072959217, 6.73052010605544557945206546422, 7.39223584461253305173428087103, 8.951433118344297101039784671225, 10.99699942212241452681758323888, 11.40576683241105103367060315179, 12.50080686695973950220311911607, 13.73683219910901594411155851118, 15.25327777972161811298757579476