L(s) = 1 | + (1.35 − 2.35i)2-s + 3-s + (−2.68 − 4.65i)4-s + (1.94 + 3.36i)5-s + (1.35 − 2.35i)6-s + (0.587 − 2.57i)7-s − 9.16·8-s + 9-s + 10.5·10-s − 1.63·11-s + (−2.68 − 4.65i)12-s + (−3.59 + 0.202i)13-s + (−5.26 − 4.88i)14-s + (1.94 + 3.36i)15-s + (−7.06 + 12.2i)16-s + (2.09 + 3.63i)17-s + ⋯ |
L(s) = 1 | + (0.960 − 1.66i)2-s + 0.577·3-s + (−1.34 − 2.32i)4-s + (0.869 + 1.50i)5-s + (0.554 − 0.960i)6-s + (0.221 − 0.975i)7-s − 3.23·8-s + 0.333·9-s + 3.33·10-s − 0.491·11-s + (−0.775 − 1.34i)12-s + (−0.998 + 0.0562i)13-s + (−1.40 − 1.30i)14-s + (0.501 + 0.869i)15-s + (−1.76 + 3.05i)16-s + (0.508 + 0.880i)17-s + ⋯ |
Λ(s)=(=(273s/2ΓC(s)L(s)(−0.467+0.884i)Λ(2−s)
Λ(s)=(=(273s/2ΓC(s+1/2)L(s)(−0.467+0.884i)Λ(1−s)
Degree: |
2 |
Conductor: |
273
= 3⋅7⋅13
|
Sign: |
−0.467+0.884i
|
Analytic conductor: |
2.17991 |
Root analytic conductor: |
1.47645 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ273(172,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 273, ( :1/2), −0.467+0.884i)
|
Particular Values
L(1) |
≈ |
1.22179−2.02774i |
L(21) |
≈ |
1.22179−2.02774i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1−T |
| 7 | 1+(−0.587+2.57i)T |
| 13 | 1+(3.59−0.202i)T |
good | 2 | 1+(−1.35+2.35i)T+(−1−1.73i)T2 |
| 5 | 1+(−1.94−3.36i)T+(−2.5+4.33i)T2 |
| 11 | 1+1.63T+11T2 |
| 17 | 1+(−2.09−3.63i)T+(−8.5+14.7i)T2 |
| 19 | 1−1.69T+19T2 |
| 23 | 1+(0.395−0.685i)T+(−11.5−19.9i)T2 |
| 29 | 1+(0.242+0.419i)T+(−14.5+25.1i)T2 |
| 31 | 1+(0.915−1.58i)T+(−15.5−26.8i)T2 |
| 37 | 1+(0.344−0.596i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−2.96−5.14i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−2.79+4.83i)T+(−21.5−37.2i)T2 |
| 47 | 1+(0.292+0.506i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−3.04+5.28i)T+(−26.5−45.8i)T2 |
| 59 | 1+(4.13+7.16i)T+(−29.5+51.0i)T2 |
| 61 | 1+9.08T+61T2 |
| 67 | 1−1.00T+67T2 |
| 71 | 1+(−7.93+13.7i)T+(−35.5−61.4i)T2 |
| 73 | 1+(2.92−5.06i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−0.643−1.11i)T+(−39.5+68.4i)T2 |
| 83 | 1+6.18T+83T2 |
| 89 | 1+(−2.20+3.81i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−5.08+8.81i)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
−11.46512591867982449722739708829, −10.53293981840280292886633379628, −10.21554684014014106923290272684, −9.434069032865662389713587898909, −7.59934297719779133503924016895, −6.33805424833727083133541641290, −5.09053288476536895500970416380, −3.74745726056237636322126896234, −2.87080539323744406561502597833, −1.83587245275733258189936941279,
2.67413685104353343404128901411, 4.50901684783034205920764220299, 5.25323736161498325572429240065, 5.85843179189135299895945690132, 7.36724996373362935893197626507, 8.184077203168998604290513607797, 9.058912337905726375807854226652, 9.586692284425390851083826903413, 12.10059380085386625976592495862, 12.52305171949881466981957092117