L(s) = 1 | + (−1.21 − 2.10i)2-s + 0.753·3-s + (−1.95 + 3.39i)4-s + (−0.170 + 0.295i)5-s + (−0.916 − 1.58i)6-s + 4.65·8-s − 2.43·9-s + 0.830·10-s − 2.43·11-s + (−1.47 + 2.55i)12-s + (2.50 + 2.59i)13-s + (−0.128 + 0.222i)15-s + (−1.74 − 3.02i)16-s + (0.974 − 1.68i)17-s + (2.95 + 5.12i)18-s − 6.29·19-s + ⋯ |
L(s) = 1 | + (−0.859 − 1.48i)2-s + 0.435·3-s + (−0.978 + 1.69i)4-s + (−0.0763 + 0.132i)5-s + (−0.374 − 0.647i)6-s + 1.64·8-s − 0.810·9-s + 0.262·10-s − 0.733·11-s + (−0.425 + 0.737i)12-s + (0.693 + 0.720i)13-s + (−0.0332 + 0.0575i)15-s + (−0.437 − 0.757i)16-s + (0.236 − 0.409i)17-s + (0.697 + 1.20i)18-s − 1.44·19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.910−0.413i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.910−0.413i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.910−0.413i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(373,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.910−0.413i)
|
Particular Values
L(1) |
≈ |
0.518830+0.112174i |
L(21) |
≈ |
0.518830+0.112174i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(−2.50−2.59i)T |
good | 2 | 1+(1.21+2.10i)T+(−1+1.73i)T2 |
| 3 | 1−0.753T+3T2 |
| 5 | 1+(0.170−0.295i)T+(−2.5−4.33i)T2 |
| 11 | 1+2.43T+11T2 |
| 17 | 1+(−0.974+1.68i)T+(−8.5−14.7i)T2 |
| 19 | 1+6.29T+19T2 |
| 23 | 1+(−1.84−3.19i)T+(−11.5+19.9i)T2 |
| 29 | 1+(2.22−3.84i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−0.987−1.71i)T+(−15.5+26.8i)T2 |
| 37 | 1+(−4.81−8.33i)T+(−18.5+32.0i)T2 |
| 41 | 1+(6.26−10.8i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−4.20−7.28i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−4.50+7.79i)T+(−23.5−40.7i)T2 |
| 53 | 1+(0.746+1.29i)T+(−26.5+45.8i)T2 |
| 59 | 1+(0.313−0.542i)T+(−29.5−51.0i)T2 |
| 61 | 1+1.14T+61T2 |
| 67 | 1+5.59T+67T2 |
| 71 | 1+(4.74+8.22i)T+(−35.5+61.4i)T2 |
| 73 | 1+(5.95+10.3i)T+(−36.5+63.2i)T2 |
| 79 | 1+(2.23−3.87i)T+(−39.5−68.4i)T2 |
| 83 | 1−1.41T+83T2 |
| 89 | 1+(6.22+10.7i)T+(−44.5+77.0i)T2 |
| 97 | 1+(5.13+8.90i)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
−10.74156196655733833986584841084, −9.820825683583443916993675815681, −8.957123227238904047699948640436, −8.481187572506083493844367524053, −7.57628010844268431990553783704, −6.21670511005601086260079659492, −4.74276953406743188170809781075, −3.43955744881771086119678571796, −2.76111739508765026665203052232, −1.53643566161148542785053123402,
0.36453354893281442051376943767, 2.54397241791588699981131003131, 4.17258647250796809360704273054, 5.57019992730040490130479764472, 6.03392700918412714462102576527, 7.16744331647070007449129894764, 8.111106345820478502727110745413, 8.501911411568756452069907564365, 9.176581945630727948690227827998, 10.38706769655740612920630871283