L(s) = 1 | + (16 + 16i)2-s + (57 − 57i)3-s + 512i·4-s + (2.92e3 + 1.10e3i)5-s + 1.82e3·6-s + (6.95e3 + 6.95e3i)7-s + (−8.19e3 + 8.19e3i)8-s + 5.25e4i·9-s + (2.92e4 + 6.44e4i)10-s + 7.52e4·11-s + (2.91e4 + 2.91e4i)12-s + (1.09e5 − 1.09e5i)13-s + 2.22e5i·14-s + (2.29e5 − 1.04e5i)15-s − 2.62e5·16-s + (−1.52e6 − 1.52e6i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.234 − 0.234i)3-s + 0.5i·4-s + (0.936 + 0.352i)5-s + 0.234·6-s + (0.413 + 0.413i)7-s + (−0.250 + 0.250i)8-s + 0.889i·9-s + (0.292 + 0.644i)10-s + 0.467·11-s + (0.117 + 0.117i)12-s + (0.295 − 0.295i)13-s + 0.413i·14-s + (0.302 − 0.136i)15-s − 0.250·16-s + (−1.07 − 1.07i)17-s + ⋯ |
Λ(s)=(=(10s/2ΓC(s)L(s)(0.557−0.830i)Λ(11−s)
Λ(s)=(=(10s/2ΓC(s+5)L(s)(0.557−0.830i)Λ(1−s)
Degree: |
2 |
Conductor: |
10
= 2⋅5
|
Sign: |
0.557−0.830i
|
Analytic conductor: |
6.35357 |
Root analytic conductor: |
2.52062 |
Motivic weight: |
10 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ10(7,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 10, ( :5), 0.557−0.830i)
|
Particular Values
L(211) |
≈ |
2.22479+1.18563i |
L(21) |
≈ |
2.22479+1.18563i |
L(6) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−16−16i)T |
| 5 | 1+(−2.92e3−1.10e3i)T |
good | 3 | 1+(−57+57i)T−5.90e4iT2 |
| 7 | 1+(−6.95e3−6.95e3i)T+2.82e8iT2 |
| 11 | 1−7.52e4T+2.59e10T2 |
| 13 | 1+(−1.09e5+1.09e5i)T−1.37e11iT2 |
| 17 | 1+(1.52e6+1.52e6i)T+2.01e12iT2 |
| 19 | 1+4.03e6iT−6.13e12T2 |
| 23 | 1+(7.12e5−7.12e5i)T−4.14e13iT2 |
| 29 | 1−4.46e5iT−4.20e14T2 |
| 31 | 1+2.90e7T+8.19e14T2 |
| 37 | 1+(9.11e5+9.11e5i)T+4.80e15iT2 |
| 41 | 1+1.63e8T+1.34e16T2 |
| 43 | 1+(−1.18e8+1.18e8i)T−2.16e16iT2 |
| 47 | 1+(−2.76e8−2.76e8i)T+5.25e16iT2 |
| 53 | 1+(−3.08e8+3.08e8i)T−1.74e17iT2 |
| 59 | 1+9.40e8iT−5.11e17T2 |
| 61 | 1+1.35e9T+7.13e17T2 |
| 67 | 1+(−8.53e8−8.53e8i)T+1.82e18iT2 |
| 71 | 1−2.82e9T+3.25e18T2 |
| 73 | 1+(2.75e9−2.75e9i)T−4.29e18iT2 |
| 79 | 1−3.32e9iT−9.46e18T2 |
| 83 | 1+(−1.34e9+1.34e9i)T−1.55e19iT2 |
| 89 | 1+2.66e9iT−3.11e19T2 |
| 97 | 1+(5.26e8+5.26e8i)T+7.37e19iT2 |
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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
−18.39634524542139402580114584633, −17.26944608037313628008584437461, −15.60467157306474261588579157493, −14.11076234856294098963190874907, −13.21320892687923712486291325465, −11.14695353300958144043870223346, −8.948689522200678052404287025130, −6.98215040703774643879951698499, −5.16464369333576426631803464446, −2.37842261372478713179066510225,
1.58099731909072381970786310110, 4.02374903709250392214806328264, 6.13260724991758774708242829790, 8.987284424356253584939278611471, 10.48187388561386472360744690374, 12.32121513427175594119994418304, 13.78833395774184528742258799633, 14.91836961386299830575491805189, 16.89361649096381474158077676476, 18.24201118133087810164147784822