L(s) = 1 | + (0.917 + 0.917i)2-s + (−1.69 − 0.703i)3-s − 0.318i·4-s + (−0.912 − 2.20i)6-s + (1.34 + 3.23i)7-s + (2.12 − 2.12i)8-s + (0.267 + 0.267i)9-s + (−1.46 − 3.53i)11-s + (−0.223 + 0.540i)12-s + 3.99·13-s + (−1.74 + 4.20i)14-s + 3.26·16-s + (2.51 − 3.26i)17-s + 0.490i·18-s + (3.98 − 3.98i)19-s + ⋯ |
L(s) = 1 | + (0.648 + 0.648i)2-s + (−0.980 − 0.406i)3-s − 0.159i·4-s + (−0.372 − 0.899i)6-s + (0.507 + 1.22i)7-s + (0.751 − 0.751i)8-s + (0.0890 + 0.0890i)9-s + (−0.441 − 1.06i)11-s + (−0.0645 + 0.155i)12-s + 1.10·13-s + (−0.465 + 1.12i)14-s + 0.815·16-s + (0.611 − 0.791i)17-s + 0.115i·18-s + (0.914 − 0.914i)19-s + ⋯ |
Λ(s)=(=(425s/2ΓC(s)L(s)(0.966+0.256i)Λ(2−s)
Λ(s)=(=(425s/2ΓC(s+1/2)L(s)(0.966+0.256i)Λ(1−s)
Degree: |
2 |
Conductor: |
425
= 52⋅17
|
Sign: |
0.966+0.256i
|
Analytic conductor: |
3.39364 |
Root analytic conductor: |
1.84218 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ425(399,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 425, ( :1/2), 0.966+0.256i)
|
Particular Values
L(1) |
≈ |
1.51849−0.198371i |
L(21) |
≈ |
1.51849−0.198371i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1+(−2.51+3.26i)T |
good | 2 | 1+(−0.917−0.917i)T+2iT2 |
| 3 | 1+(1.69+0.703i)T+(2.12+2.12i)T2 |
| 7 | 1+(−1.34−3.23i)T+(−4.94+4.94i)T2 |
| 11 | 1+(1.46+3.53i)T+(−7.77+7.77i)T2 |
| 13 | 1−3.99T+13T2 |
| 19 | 1+(−3.98+3.98i)T−19iT2 |
| 23 | 1+(2.31−0.960i)T+(16.2−16.2i)T2 |
| 29 | 1+(−5.14−2.12i)T+(20.5+20.5i)T2 |
| 31 | 1+(−0.843+2.03i)T+(−21.9−21.9i)T2 |
| 37 | 1+(1.78+0.738i)T+(26.1+26.1i)T2 |
| 41 | 1+(−0.730+0.302i)T+(28.9−28.9i)T2 |
| 43 | 1+(0.704−0.704i)T−43iT2 |
| 47 | 1+9.47T+47T2 |
| 53 | 1+(−3.87−3.87i)T+53iT2 |
| 59 | 1+(2.12+2.12i)T+59iT2 |
| 61 | 1+(12.9−5.37i)T+(43.1−43.1i)T2 |
| 67 | 1−15.9iT−67T2 |
| 71 | 1+(2.09−5.05i)T+(−50.2−50.2i)T2 |
| 73 | 1+(−2.51+6.06i)T+(−51.6−51.6i)T2 |
| 79 | 1+(−4.54−10.9i)T+(−55.8+55.8i)T2 |
| 83 | 1+(10.8+10.8i)T+83iT2 |
| 89 | 1+4.92iT−89T2 |
| 97 | 1+(5.45−13.1i)T+(−68.5−68.5i)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
−11.44108254750162680389589301203, −10.51011601853258556472708642573, −9.210970143951580637856538420917, −8.258601369760602607877352529715, −7.06734631593927198039367865954, −6.01669727344474931532233097160, −5.64157435812079741264051642404, −4.86584022223957655553647999204, −3.11346120166255624884036300835, −1.06799412543329883342493773436,
1.57497881900485251946114063693, 3.43646001515816764822894419101, 4.37161194771167583268174266613, 5.08480855459943121032757760288, 6.25316027219527117612485628550, 7.62817438683023094113087115598, 8.212896208170197532738940381994, 10.09405987479042656674953283167, 10.47951175107490237945124433878, 11.24981115012732815385785236781