L(s) = 1 | + (0.5 + 0.866i)2-s + (0.278 + 0.0746i)3-s + (−0.499 + 0.866i)4-s + (0.0746 + 0.278i)6-s + (−4.15 − 2.39i)7-s − 0.999·8-s + (−2.52 − 1.45i)9-s + (−0.105 + 0.395i)11-s + (−0.203 + 0.203i)12-s + (2.47 − 2.61i)13-s − 4.79i·14-s + (−0.5 − 0.866i)16-s + (−0.999 − 3.73i)17-s − 2.91i·18-s + (5.05 − 1.35i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.160 + 0.0430i)3-s + (−0.249 + 0.433i)4-s + (0.0304 + 0.113i)6-s + (−1.56 − 0.905i)7-s − 0.353·8-s + (−0.842 − 0.486i)9-s + (−0.0319 + 0.119i)11-s + (−0.0588 + 0.0588i)12-s + (0.687 − 0.726i)13-s − 1.28i·14-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.904i)17-s − 0.687i·18-s + (1.16 − 0.311i)19-s + ⋯ |
Λ(s)=(=(650s/2ΓC(s)L(s)(0.191+0.981i)Λ(2−s)
Λ(s)=(=(650s/2ΓC(s+1/2)L(s)(0.191+0.981i)Λ(1−s)
Degree: |
2 |
Conductor: |
650
= 2⋅52⋅13
|
Sign: |
0.191+0.981i
|
Analytic conductor: |
5.19027 |
Root analytic conductor: |
2.27821 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ650(457,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 650, ( :1/2), 0.191+0.981i)
|
Particular Values
L(1) |
≈ |
0.679025−0.559303i |
L(21) |
≈ |
0.679025−0.559303i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5−0.866i)T |
| 5 | 1 |
| 13 | 1+(−2.47+2.61i)T |
good | 3 | 1+(−0.278−0.0746i)T+(2.59+1.5i)T2 |
| 7 | 1+(4.15+2.39i)T+(3.5+6.06i)T2 |
| 11 | 1+(0.105−0.395i)T+(−9.52−5.5i)T2 |
| 17 | 1+(0.999+3.73i)T+(−14.7+8.5i)T2 |
| 19 | 1+(−5.05+1.35i)T+(16.4−9.5i)T2 |
| 23 | 1+(−1.60+5.97i)T+(−19.9−11.5i)T2 |
| 29 | 1+(8.13−4.69i)T+(14.5−25.1i)T2 |
| 31 | 1+(−0.0701+0.0701i)T−31iT2 |
| 37 | 1+(9.07−5.23i)T+(18.5−32.0i)T2 |
| 41 | 1+(−5.48−1.46i)T+(35.5+20.5i)T2 |
| 43 | 1+(7.42−1.99i)T+(37.2−21.5i)T2 |
| 47 | 1+6.95iT−47T2 |
| 53 | 1+(3.07−3.07i)T−53iT2 |
| 59 | 1+(0.564+2.10i)T+(−51.0+29.5i)T2 |
| 61 | 1+(−2.13+3.69i)T+(−30.5−52.8i)T2 |
| 67 | 1+(0.986+1.70i)T+(−33.5+58.0i)T2 |
| 71 | 1+(2.91+10.8i)T+(−61.4+35.5i)T2 |
| 73 | 1+0.776T+73T2 |
| 79 | 1+1.19iT−79T2 |
| 83 | 1+11.3iT−83T2 |
| 89 | 1+(3.47+0.931i)T+(77.0+44.5i)T2 |
| 97 | 1+(−6.51+11.2i)T+(−48.5−84.0i)T2 |
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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
−10.23329005956937307840956115106, −9.366621199095562573500244766633, −8.685046253128207211038366834966, −7.47091031392166716053359413409, −6.77190589013837156872711960017, −5.99385263966694402816170455613, −4.93683390533176760789601064302, −3.46648162586341988856010773583, −3.12480466314687537899488802600, −0.39569283310829816424964694910,
1.94231610980297535279475997899, 3.13596414684201759744445861774, 3.80237019810686282491156481225, 5.56925514692140970804413941175, 5.86981125912721670034526491652, 7.08760917441680299501097122897, 8.412797240508871416571926766876, 9.231438066645538298713592033624, 9.732041655092254422269740158106, 10.91671216701489436011318268811