L(s) = 1 | + (−17.0 + 9.84i)2-s + (−12.0 − 45.1i)3-s + (130. − 225. i)4-s + (147. + 255. i)5-s + (651. + 651. i)6-s + (−45.1 − 906. i)7-s + 2.60e3i·8-s + (−1.89e3 + 1.09e3i)9-s + (−5.03e3 − 2.90e3i)10-s + (−2.53e3 − 1.46e3i)11-s + (−1.17e4 − 3.15e3i)12-s + 1.24e4i·13-s + (9.69e3 + 1.50e4i)14-s + (9.76e3 − 9.75e3i)15-s + (−8.97e3 − 1.55e4i)16-s + (−1.37e4 + 2.38e4i)17-s + ⋯ |
L(s) = 1 | + (−1.50 + 0.870i)2-s + (−0.258 − 0.966i)3-s + (1.01 − 1.75i)4-s + (0.528 + 0.914i)5-s + (1.23 + 1.23i)6-s + (−0.0497 − 0.998i)7-s + 1.79i·8-s + (−0.866 + 0.499i)9-s + (−1.59 − 0.919i)10-s + (−0.573 − 0.330i)11-s + (−1.96 − 0.526i)12-s + 1.57i·13-s + (0.944 + 1.46i)14-s + (0.746 − 0.746i)15-s + (−0.547 − 0.949i)16-s + (−0.678 + 1.17i)17-s + ⋯ |
Λ(s)=(=(21s/2ΓC(s)L(s)(−0.820−0.572i)Λ(8−s)
Λ(s)=(=(21s/2ΓC(s+7/2)L(s)(−0.820−0.572i)Λ(1−s)
Degree: |
2 |
Conductor: |
21
= 3⋅7
|
Sign: |
−0.820−0.572i
|
Analytic conductor: |
6.56008 |
Root analytic conductor: |
2.56126 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ21(17,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 21, ( :7/2), −0.820−0.572i)
|
Particular Values
L(4) |
≈ |
0.0948180+0.301615i |
L(21) |
≈ |
0.0948180+0.301615i |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(12.0+45.1i)T |
| 7 | 1+(45.1+906.i)T |
good | 2 | 1+(17.0−9.84i)T+(64−110.i)T2 |
| 5 | 1+(−147.−255.i)T+(−3.90e4+6.76e4i)T2 |
| 11 | 1+(2.53e3+1.46e3i)T+(9.74e6+1.68e7i)T2 |
| 13 | 1−1.24e4iT−6.27e7T2 |
| 17 | 1+(1.37e4−2.38e4i)T+(−2.05e8−3.55e8i)T2 |
| 19 | 1+(3.84e3−2.21e3i)T+(4.46e8−7.74e8i)T2 |
| 23 | 1+(3.40e4−1.96e4i)T+(1.70e9−2.94e9i)T2 |
| 29 | 1−1.04e5iT−1.72e10T2 |
| 31 | 1+(3.49e3+2.02e3i)T+(1.37e10+2.38e10i)T2 |
| 37 | 1+(−1.88e5−3.26e5i)T+(−4.74e10+8.22e10i)T2 |
| 41 | 1+2.34e5T+1.94e11T2 |
| 43 | 1+6.55e5T+2.71e11T2 |
| 47 | 1+(4.50e4+7.80e4i)T+(−2.53e11+4.38e11i)T2 |
| 53 | 1+(−4.01e5−2.31e5i)T+(5.87e11+1.01e12i)T2 |
| 59 | 1+(−1.12e5+1.94e5i)T+(−1.24e12−2.15e12i)T2 |
| 61 | 1+(−2.12e6+1.22e6i)T+(1.57e12−2.72e12i)T2 |
| 67 | 1+(2.21e6−3.82e6i)T+(−3.03e12−5.24e12i)T2 |
| 71 | 1+4.85e6iT−9.09e12T2 |
| 73 | 1+(−1.42e6−8.20e5i)T+(5.52e12+9.56e12i)T2 |
| 79 | 1+(5.86e5+1.01e6i)T+(−9.60e12+1.66e13i)T2 |
| 83 | 1+7.03e6T+2.71e13T2 |
| 89 | 1+(2.11e5+3.67e5i)T+(−2.21e13+3.83e13i)T2 |
| 97 | 1+4.84e6iT−8.07e13T2 |
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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
−17.24138593156365052475302899232, −16.40545679227864789327305459680, −14.63030003913393118872991238715, −13.51331798146089750324820387738, −11.16807973118008463044189723636, −10.15172495215270272239098639911, −8.391362574289126604448769619411, −7.02052528806474892105472095740, −6.32690821001983921277784695141, −1.69753762143710169504592749484,
0.28595968277332979834181896463, 2.60005455964037568351486951287, 5.32539932262742369394584451183, 8.305101650520161030511797785269, 9.319026777559085958297394248137, 10.22414829544985341380687092607, 11.59356743341455053436087511161, 12.82152692878901315608294919232, 15.39838545800286061620551721972, 16.37798028972311270817207601579