L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 − 0.866i)4-s + (2 − 3.46i)5-s + 3·6-s + 0.999·8-s + (−3 + 5.19i)9-s + (1.99 + 3.46i)10-s + (−0.5 − 0.866i)11-s + (−1.50 + 2.59i)12-s − 13-s − 12·15-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)18-s + (3 − 5.19i)19-s − 3.99·20-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.894 − 1.54i)5-s + 1.22·6-s + 0.353·8-s + (−1 + 1.73i)9-s + (0.632 + 1.09i)10-s + (−0.150 − 0.261i)11-s + (−0.433 + 0.749i)12-s − 0.277·13-s − 3.09·15-s + (−0.125 + 0.216i)16-s + (−0.707 − 1.22i)18-s + (0.688 − 1.19i)19-s − 0.894·20-s + ⋯ |
Λ(s)=(=(1274s/2ΓC(s)L(s)(−0.991−0.126i)Λ(2−s)
Λ(s)=(=(1274s/2ΓC(s+1/2)L(s)(−0.991−0.126i)Λ(1−s)
Degree: |
2 |
Conductor: |
1274
= 2⋅72⋅13
|
Sign: |
−0.991−0.126i
|
Analytic conductor: |
10.1729 |
Root analytic conductor: |
3.18950 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1274(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1274, ( :1/2), −0.991−0.126i)
|
Particular Values
L(1) |
≈ |
0.8070926367 |
L(21) |
≈ |
0.8070926367 |
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 |
| 7 | 1 |
| 13 | 1+T |
good | 3 | 1+(1.5+2.59i)T+(−1.5+2.59i)T2 |
| 5 | 1+(−2+3.46i)T+(−2.5−4.33i)T2 |
| 11 | 1+(0.5+0.866i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−8.5+14.7i)T2 |
| 19 | 1+(−3+5.19i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−3.5+6.06i)T+(−11.5−19.9i)T2 |
| 29 | 1+4T+29T2 |
| 31 | 1+(3.5+6.06i)T+(−15.5+26.8i)T2 |
| 37 | 1+(4.5−7.79i)T+(−18.5−32.0i)T2 |
| 41 | 1+3T+41T2 |
| 43 | 1−4T+43T2 |
| 47 | 1+(3.5−6.06i)T+(−23.5−40.7i)T2 |
| 53 | 1+(−26.5+45.8i)T2 |
| 59 | 1+(−5−8.66i)T+(−29.5+51.0i)T2 |
| 61 | 1+(0.5−0.866i)T+(−30.5−52.8i)T2 |
| 67 | 1+(0.5+0.866i)T+(−33.5+58.0i)T2 |
| 71 | 1−16T+71T2 |
| 73 | 1+(2.5+4.33i)T+(−36.5+63.2i)T2 |
| 79 | 1+(5.5−9.52i)T+(−39.5−68.4i)T2 |
| 83 | 1+83T2 |
| 89 | 1+(−3+5.19i)T+(−44.5−77.0i)T2 |
| 97 | 1+T+97T2 |
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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
−9.021594127433278379638206563697, −8.382568948786264269487885069334, −7.52797768626457675565164781243, −6.73565909885860207790899875734, −5.96928570800475926006589990578, −5.28574528877152935914182696175, −4.71016423673237050421320595122, −2.37640809178145175169787267256, −1.28424668194566112350330942710, −0.45543412837843330237426882154,
1.92614253494612465849687376707, 3.30293151351504200317268486616, 3.70039284966342558337052914706, 5.21295223087484650788380698945, 5.62392069734430319538721511653, 6.74344230799443447443386479398, 7.55132031901874429827242824087, 9.051353354120912711582417601800, 9.691263019391763073838050297185, 10.13371298581512180636247451432