L(s) = 1 | + (−1.24 − 0.672i)2-s + (1.34 − 0.555i)3-s + (1.09 + 1.67i)4-s + (1.54 + 3.73i)5-s + (−2.04 − 0.211i)6-s + 0.167i·7-s + (−0.234 − 2.81i)8-s + (−0.632 + 0.632i)9-s + (0.589 − 5.68i)10-s + (0.115 − 0.0478i)11-s + (2.39 + 1.63i)12-s + (0.804 + 3.51i)13-s + (0.112 − 0.208i)14-s + (4.14 + 4.14i)15-s + (−1.60 + 3.66i)16-s − 4.75·17-s + ⋯ |
L(s) = 1 | + (−0.879 − 0.475i)2-s + (0.773 − 0.320i)3-s + (0.547 + 0.837i)4-s + (0.691 + 1.67i)5-s + (−0.833 − 0.0863i)6-s + 0.0632i·7-s + (−0.0829 − 0.996i)8-s + (−0.210 + 0.210i)9-s + (0.186 − 1.79i)10-s + (0.0348 − 0.0144i)11-s + (0.691 + 0.472i)12-s + (0.223 + 0.974i)13-s + (0.0301 − 0.0556i)14-s + (1.07 + 1.07i)15-s + (−0.401 + 0.915i)16-s − 1.15·17-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(0.704−0.709i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(0.704−0.709i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
0.704−0.709i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(411,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), 0.704−0.709i)
|
Particular Values
L(1) |
≈ |
1.13388+0.472203i |
L(21) |
≈ |
1.13388+0.472203i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.24+0.672i)T |
| 13 | 1+(−0.804−3.51i)T |
good | 3 | 1+(−1.34+0.555i)T+(2.12−2.12i)T2 |
| 5 | 1+(−1.54−3.73i)T+(−3.53+3.53i)T2 |
| 7 | 1−0.167iT−7T2 |
| 11 | 1+(−0.115+0.0478i)T+(7.77−7.77i)T2 |
| 17 | 1+4.75T+17T2 |
| 19 | 1+(−1.08+2.62i)T+(−13.4−13.4i)T2 |
| 23 | 1+(0.822−0.822i)T−23iT2 |
| 29 | 1+(0.362+0.876i)T+(−20.5+20.5i)T2 |
| 31 | 1+(−1.60+1.60i)T−31iT2 |
| 37 | 1+(−9.94+4.11i)T+(26.1−26.1i)T2 |
| 41 | 1−1.57T+41T2 |
| 43 | 1+(−2.99+7.23i)T+(−30.4−30.4i)T2 |
| 47 | 1+(−5.14+5.14i)T−47iT2 |
| 53 | 1+(−0.952+2.29i)T+(−37.4−37.4i)T2 |
| 59 | 1+(−4.80−11.6i)T+(−41.7+41.7i)T2 |
| 61 | 1+(1.39+3.37i)T+(−43.1+43.1i)T2 |
| 67 | 1+(−5.27−2.18i)T+(47.3+47.3i)T2 |
| 71 | 1+12.4T+71T2 |
| 73 | 1+8.93iT−73T2 |
| 79 | 1−14.3T+79T2 |
| 83 | 1+(−4.24+10.2i)T+(−58.6−58.6i)T2 |
| 89 | 1−6.33T+89T2 |
| 97 | 1+(7.29+7.29i)T+97iT2 |
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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
−11.08839064998793446157461530321, −10.49156991744301373802282762928, −9.395764278324130480670248586605, −8.845259520242848239469663506617, −7.58074702816235839169922247761, −6.97811199170468831583591738910, −6.06735388008582330289790731998, −3.87786080962000735280154133621, −2.59577277094705939570397256442, −2.15837838765190770317265431945,
0.989540250095804174640263152590, 2.50400782621931479362899056075, 4.35277405225942873493620876199, 5.49830973588569366581058052535, 6.29745707851370418007694637281, 7.954155468348914991594762960124, 8.409746869841275489189698094703, 9.274229733122500580553914362311, 9.665345619885795411471800111371, 10.76423921269075994541411564384