L(s) = 1 | + (0.292 − 0.292i)2-s + (1 + 2.41i)3-s + 1.82i·4-s + (1 + 0.414i)6-s + (1 + 0.414i)7-s + (1.12 + 1.12i)8-s + (−2.70 + 2.70i)9-s + (−1 − 0.414i)11-s + (−4.41 + 1.82i)12-s + 1.41·13-s + (0.414 − 0.171i)14-s − 3·16-s + (3 − 2.82i)17-s + 1.58i·18-s + (−3.41 − 3.41i)19-s + ⋯ |
L(s) = 1 | + (0.207 − 0.207i)2-s + (0.577 + 1.39i)3-s + 0.914i·4-s + (0.408 + 0.169i)6-s + (0.377 + 0.156i)7-s + (0.396 + 0.396i)8-s + (−0.902 + 0.902i)9-s + (−0.301 − 0.124i)11-s + (−1.27 + 0.527i)12-s + 0.392·13-s + (0.110 − 0.0458i)14-s − 0.750·16-s + (0.727 − 0.685i)17-s + 0.373i·18-s + (−0.783 − 0.783i)19-s + ⋯ |
Λ(s)=(=(425s/2ΓC(s)L(s)(−0.243−0.969i)Λ(2−s)
Λ(s)=(=(425s/2ΓC(s+1/2)L(s)(−0.243−0.969i)Λ(1−s)
Degree: |
2 |
Conductor: |
425
= 52⋅17
|
Sign: |
−0.243−0.969i
|
Analytic conductor: |
3.39364 |
Root analytic conductor: |
1.84218 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ425(274,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 425, ( :1/2), −0.243−0.969i)
|
Particular Values
L(1) |
≈ |
1.15471+1.48017i |
L(21) |
≈ |
1.15471+1.48017i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1+(−3+2.82i)T |
good | 2 | 1+(−0.292+0.292i)T−2iT2 |
| 3 | 1+(−1−2.41i)T+(−2.12+2.12i)T2 |
| 7 | 1+(−1−0.414i)T+(4.94+4.94i)T2 |
| 11 | 1+(1+0.414i)T+(7.77+7.77i)T2 |
| 13 | 1−1.41T+13T2 |
| 19 | 1+(3.41+3.41i)T+19iT2 |
| 23 | 1+(−1.58+3.82i)T+(−16.2−16.2i)T2 |
| 29 | 1+(−1.70−4.12i)T+(−20.5+20.5i)T2 |
| 31 | 1+(3−1.24i)T+(21.9−21.9i)T2 |
| 37 | 1+(1.46+3.53i)T+(−26.1+26.1i)T2 |
| 41 | 1+(3.12−7.53i)T+(−28.9−28.9i)T2 |
| 43 | 1+(−3.41−3.41i)T+43iT2 |
| 47 | 1−10.8T+47T2 |
| 53 | 1+(1−i)T−53iT2 |
| 59 | 1+(−4.24+4.24i)T−59iT2 |
| 61 | 1+(−3.53+8.53i)T+(−43.1−43.1i)T2 |
| 67 | 1−6.82iT−67T2 |
| 71 | 1+(−12.0+5i)T+(50.2−50.2i)T2 |
| 73 | 1+(4.94−2.05i)T+(51.6−51.6i)T2 |
| 79 | 1+(−3.82−1.58i)T+(55.8+55.8i)T2 |
| 83 | 1+(0.242−0.242i)T−83iT2 |
| 89 | 1+9.41iT−89T2 |
| 97 | 1+(5.94−2.46i)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
−11.17834442829990635821476442828, −10.69279241486004725532513321552, −9.546067370968706705099870425912, −8.732314298738925788404848615859, −8.134206742415446450192966580123, −6.91273253948972901670792519503, −5.21450512299763379390149636949, −4.45951147905015743018339951142, −3.46139040227468367595713524718, −2.57639039715506029730153934450,
1.20121589693157902927855301237, 2.19900918266107781696633965872, 3.92187076162873379641938516713, 5.44298970401391964287553690734, 6.24854746752052352403603705543, 7.21833540647213711024846349012, 7.975986757986551377851968997761, 8.874200394756401818404148840388, 10.09277762495400667169469511826, 10.88011837613840512510787890070