L(s) = 1 | + (1.20 + 2.09i)3-s + 2.82·5-s + (−2.20 + 3.82i)7-s + (−1.41 + 2.44i)9-s + (−1.62 − 2.80i)11-s + (−1 + 3.46i)13-s + (3.41 + 5.91i)15-s + (2.91 − 5.04i)17-s + (0.621 − 1.07i)19-s − 10.6·21-s + (−0.621 − 1.07i)23-s + 3.00·25-s + 0.414·27-s + (−4.32 − 7.49i)29-s + 5.65·31-s + ⋯ |
L(s) = 1 | + (0.696 + 1.20i)3-s + 1.26·5-s + (−0.834 + 1.44i)7-s + (−0.471 + 0.816i)9-s + (−0.488 − 0.846i)11-s + (−0.277 + 0.960i)13-s + (0.881 + 1.52i)15-s + (0.706 − 1.22i)17-s + (0.142 − 0.246i)19-s − 2.32·21-s + (−0.129 − 0.224i)23-s + 0.600·25-s + 0.0797·27-s + (−0.803 − 1.39i)29-s + 1.01·31-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(0.0128−0.999i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(0.0128−0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
0.0128−0.999i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(289,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), 0.0128−0.999i)
|
Particular Values
L(1) |
≈ |
1.33304+1.31606i |
L(21) |
≈ |
1.33304+1.31606i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1+(1−3.46i)T |
good | 3 | 1+(−1.20−2.09i)T+(−1.5+2.59i)T2 |
| 5 | 1−2.82T+5T2 |
| 7 | 1+(2.20−3.82i)T+(−3.5−6.06i)T2 |
| 11 | 1+(1.62+2.80i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−2.91+5.04i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−0.621+1.07i)T+(−9.5−16.4i)T2 |
| 23 | 1+(0.621+1.07i)T+(−11.5+19.9i)T2 |
| 29 | 1+(4.32+7.49i)T+(−14.5+25.1i)T2 |
| 31 | 1−5.65T+31T2 |
| 37 | 1+(−3.74−6.48i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−2.91−5.04i)T+(−20.5+35.5i)T2 |
| 43 | 1+(2.03−3.52i)T+(−21.5−37.2i)T2 |
| 47 | 1−6T+47T2 |
| 53 | 1+2.82T+53T2 |
| 59 | 1+(−0.621+1.07i)T+(−29.5−51.0i)T2 |
| 61 | 1+(3.5−6.06i)T+(−30.5−52.8i)T2 |
| 67 | 1+(6.62+11.4i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−3.62+6.27i)T+(−35.5−61.4i)T2 |
| 73 | 1−12.4T+73T2 |
| 79 | 1+6T+79T2 |
| 83 | 1−4T+83T2 |
| 89 | 1+(−1.67−2.89i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−4.5+7.79i)T+(−48.5−84.0i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.35484796692688337527970726096, −10.04944543107678862040808280419, −9.488616738015534926295207752071, −9.217563179927615778971584576186, −8.122432780260744564128465704578, −6.39630901918497497806049190205, −5.67336017977472620843637260124, −4.67384186769032496677940069073, −3.07031960212961596442441316760, −2.47081857615985642349475466340,
1.25426686852587588783194986356, 2.44918310669262376538627162911, 3.70892945981940961946842011871, 5.45303269820738617530664008474, 6.46960408175150434653360802389, 7.35168825284763161452944805537, 7.903609832611024544152735507094, 9.246555227633315378718886901213, 10.22384562239514511752953407474, 10.50893836827839743915766333553