L(s) = 1 | + (−43.0 + 13.9i)2-s + (296. + 215. i)3-s + (1.65e3 − 1.20e3i)4-s + (8.35e3 − 2.57e4i)5-s + (−1.57e4 − 5.13e3i)6-s + (−2.52e4 − 3.47e4i)7-s + (−5.44e4 + 7.49e4i)8-s + (−1.22e5 − 3.77e5i)9-s + 1.22e6i·10-s + (−5.79e5 + 1.67e6i)11-s + 7.51e5·12-s + (5.13e5 − 1.66e5i)13-s + (1.57e6 + 1.14e6i)14-s + (8.02e6 − 5.83e6i)15-s + (1.29e6 − 3.98e6i)16-s + (−1.06e7 − 3.46e6i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.407 + 0.295i)3-s + (0.404 − 0.293i)4-s + (0.534 − 1.64i)5-s + (−0.338 − 0.109i)6-s + (−0.214 − 0.295i)7-s + (−0.207 + 0.286i)8-s + (−0.230 − 0.710i)9-s + 1.22i·10-s + (−0.327 + 0.944i)11-s + 0.251·12-s + (0.106 − 0.0345i)13-s + (0.208 + 0.151i)14-s + (0.704 − 0.512i)15-s + (0.0772 − 0.237i)16-s + (−0.442 − 0.143i)17-s + ⋯ |
Λ(s)=(=(22s/2ΓC(s)L(s)(−0.807+0.590i)Λ(13−s)
Λ(s)=(=(22s/2ΓC(s+6)L(s)(−0.807+0.590i)Λ(1−s)
Degree: |
2 |
Conductor: |
22
= 2⋅11
|
Sign: |
−0.807+0.590i
|
Analytic conductor: |
20.1078 |
Root analytic conductor: |
4.48417 |
Motivic weight: |
12 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ22(17,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 22, ( :6), −0.807+0.590i)
|
Particular Values
L(213) |
≈ |
0.251034−0.768998i |
L(21) |
≈ |
0.251034−0.768998i |
L(7) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(43.0−13.9i)T |
| 11 | 1+(5.79e5−1.67e6i)T |
good | 3 | 1+(−296.−215.i)T+(1.64e5+5.05e5i)T2 |
| 5 | 1+(−8.35e3+2.57e4i)T+(−1.97e8−1.43e8i)T2 |
| 7 | 1+(2.52e4+3.47e4i)T+(−4.27e9+1.31e10i)T2 |
| 13 | 1+(−5.13e5+1.66e5i)T+(1.88e13−1.36e13i)T2 |
| 17 | 1+(1.06e7+3.46e6i)T+(4.71e14+3.42e14i)T2 |
| 19 | 1+(3.35e7−4.61e7i)T+(−6.83e14−2.10e15i)T2 |
| 23 | 1+1.15e8T+2.19e16T2 |
| 29 | 1+(5.32e8+7.33e8i)T+(−1.09e17+3.36e17i)T2 |
| 31 | 1+(−4.73e7−1.45e8i)T+(−6.37e17+4.62e17i)T2 |
| 37 | 1+(−2.13e9+1.55e9i)T+(2.03e18−6.26e18i)T2 |
| 41 | 1+(1.96e9−2.70e9i)T+(−6.97e18−2.14e19i)T2 |
| 43 | 1−3.28e9iT−3.99e19T2 |
| 47 | 1+(−5.64e9−4.10e9i)T+(3.59e19+1.10e20i)T2 |
| 53 | 1+(9.78e9+3.01e10i)T+(−3.97e20+2.88e20i)T2 |
| 59 | 1+(4.80e10−3.48e10i)T+(5.49e20−1.69e21i)T2 |
| 61 | 1+(−3.49e10−1.13e10i)T+(2.14e21+1.56e21i)T2 |
| 67 | 1+1.46e11T+8.18e21T2 |
| 71 | 1+(−6.00e10+1.84e11i)T+(−1.32e22−9.64e21i)T2 |
| 73 | 1+(3.40e10+4.68e10i)T+(−7.07e21+2.17e22i)T2 |
| 79 | 1+(3.22e11−1.04e11i)T+(4.78e22−3.47e22i)T2 |
| 83 | 1+(−7.18e10−2.33e10i)T+(8.64e22+6.28e22i)T2 |
| 89 | 1−6.46e11T+2.46e23T2 |
| 97 | 1+(−4.31e11−1.32e12i)T+(−5.61e23+4.07e23i)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
−14.87506973336209669202074081819, −13.26134642495403325721127789226, −12.10780140480824980530096140863, −9.992660733334584421761013891557, −9.190747322211950402038886721442, −8.029312550821049401321671525133, −6.00657976831686418842905286797, −4.31271468356864544670733489299, −1.87077108139337935114144751838, −0.31937944388891995245608200814,
2.13694399037144127741965255697, 3.06120496775943806148073082170, 6.09611777322645046776199906198, 7.39010644354115328762151952677, 8.847986879816250884839703464694, 10.47952026178811369184814301912, 11.19547799911218986724384242749, 13.25911329765158068956591467034, 14.30590891234280563920573617123, 15.61361461026954024693995821503