L(s) = 1 | + (−0.717 − 1.21i)2-s + (0.172 − 0.415i)3-s + (−0.969 + 1.74i)4-s + (−0.252 + 0.610i)5-s + (−0.629 + 0.0885i)6-s − 3.40·7-s + (2.82 − 0.0751i)8-s + (1.97 + 1.97i)9-s + (0.925 − 0.130i)10-s + (−1.70 − 0.704i)11-s + (0.560 + 0.703i)12-s + (3.28 + 1.47i)13-s + (2.44 + 4.14i)14-s + (0.210 + 0.210i)15-s + (−2.12 − 3.39i)16-s + 2.30·17-s + ⋯ |
L(s) = 1 | + (−0.507 − 0.861i)2-s + (0.0993 − 0.239i)3-s + (−0.484 + 0.874i)4-s + (−0.113 + 0.272i)5-s + (−0.257 + 0.0361i)6-s − 1.28·7-s + (0.999 − 0.0265i)8-s + (0.659 + 0.659i)9-s + (0.292 − 0.0411i)10-s + (−0.512 − 0.212i)11-s + (0.161 + 0.203i)12-s + (0.912 + 0.410i)13-s + (0.652 + 1.10i)14-s + (0.0542 + 0.0542i)15-s + (−0.530 − 0.847i)16-s + 0.559·17-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(0.984−0.177i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(0.984−0.177i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
0.984−0.177i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(99,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), 0.984−0.177i)
|
Particular Values
L(1) |
≈ |
0.877264+0.0785839i |
L(21) |
≈ |
0.877264+0.0785839i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.717+1.21i)T |
| 13 | 1+(−3.28−1.47i)T |
good | 3 | 1+(−0.172+0.415i)T+(−2.12−2.12i)T2 |
| 5 | 1+(0.252−0.610i)T+(−3.53−3.53i)T2 |
| 7 | 1+3.40T+7T2 |
| 11 | 1+(1.70+0.704i)T+(7.77+7.77i)T2 |
| 17 | 1−2.30T+17T2 |
| 19 | 1+(−2.92−7.05i)T+(−13.4+13.4i)T2 |
| 23 | 1+(−1.96−1.96i)T+23iT2 |
| 29 | 1+(−0.820−0.339i)T+(20.5+20.5i)T2 |
| 31 | 1+(1.78−1.78i)T−31iT2 |
| 37 | 1+(6.76+2.80i)T+(26.1+26.1i)T2 |
| 41 | 1−5.14iT−41T2 |
| 43 | 1+(−0.493+0.204i)T+(30.4−30.4i)T2 |
| 47 | 1+(−1.59+1.59i)T−47iT2 |
| 53 | 1+(−5.87+2.43i)T+(37.4−37.4i)T2 |
| 59 | 1+(−2.54+6.15i)T+(−41.7−41.7i)T2 |
| 61 | 1+(5.74+2.37i)T+(43.1+43.1i)T2 |
| 67 | 1+(2.69−1.11i)T+(47.3−47.3i)T2 |
| 71 | 1−7.20iT−71T2 |
| 73 | 1−6.22T+73T2 |
| 79 | 1+16.4T+79T2 |
| 83 | 1+(−0.384−0.929i)T+(−58.6+58.6i)T2 |
| 89 | 1−9.50iT−89T2 |
| 97 | 1+(8.56+8.56i)T+97iT2 |
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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
−11.06169802841131362449239928102, −10.29169314916517795929867509054, −9.673991436024970400872308925928, −8.614418185004196079466429694811, −7.65644860739699845483178208165, −6.82469844329005342880997745907, −5.42637643006964263883042536029, −3.80713995409934219012046779173, −3.06842454212734969433620097236, −1.48570281467204913465669196117,
0.74365433081774179191023450742, 3.15146565932193268372512634530, 4.46697846270765032087950026493, 5.62681198963195930626749384267, 6.66393708526679060106879105453, 7.30149667870479653845565627372, 8.626567421799399314674416570286, 9.235355794820571090033356016303, 10.07387129274528481152397085944, 10.75509591270612247399470493642