L(s) = 1 | + (−1.52 + 1.52i)2-s + (−1.06 − 2.56i)3-s − 2.67i·4-s + (5.54 + 2.29i)6-s + (−2.90 − 1.20i)7-s + (1.03 + 1.03i)8-s + (−3.32 + 3.32i)9-s + (−4.70 − 1.94i)11-s + (−6.86 + 2.84i)12-s + 1.39·13-s + (6.27 − 2.60i)14-s + 2.18·16-s + (3.66 + 1.88i)17-s − 10.1i·18-s + (3.63 + 3.63i)19-s + ⋯ |
L(s) = 1 | + (−1.08 + 1.08i)2-s + (−0.613 − 1.48i)3-s − 1.33i·4-s + (2.26 + 0.937i)6-s + (−1.09 − 0.454i)7-s + (0.366 + 0.366i)8-s + (−1.10 + 1.10i)9-s + (−1.41 − 0.587i)11-s + (−1.98 + 0.820i)12-s + 0.386·13-s + (1.67 − 0.695i)14-s + 0.545·16-s + (0.889 + 0.457i)17-s − 2.39i·18-s + (0.833 + 0.833i)19-s + ⋯ |
Λ(s)=(=(425s/2ΓC(s)L(s)(−0.552−0.833i)Λ(2−s)
Λ(s)=(=(425s/2ΓC(s+1/2)L(s)(−0.552−0.833i)Λ(1−s)
Degree: |
2 |
Conductor: |
425
= 52⋅17
|
Sign: |
−0.552−0.833i
|
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), −0.552−0.833i)
|
Particular Values
L(1) |
≈ |
0.0682320+0.127098i |
L(21) |
≈ |
0.0682320+0.127098i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1+(−3.66−1.88i)T |
good | 2 | 1+(1.52−1.52i)T−2iT2 |
| 3 | 1+(1.06+2.56i)T+(−2.12+2.12i)T2 |
| 7 | 1+(2.90+1.20i)T+(4.94+4.94i)T2 |
| 11 | 1+(4.70+1.94i)T+(7.77+7.77i)T2 |
| 13 | 1−1.39T+13T2 |
| 19 | 1+(−3.63−3.63i)T+19iT2 |
| 23 | 1+(2.37−5.73i)T+(−16.2−16.2i)T2 |
| 29 | 1+(2.65+6.41i)T+(−20.5+20.5i)T2 |
| 31 | 1+(−2.33+0.966i)T+(21.9−21.9i)T2 |
| 37 | 1+(−2.97−7.17i)T+(−26.1+26.1i)T2 |
| 41 | 1+(3.83−9.25i)T+(−28.9−28.9i)T2 |
| 43 | 1+(2.08+2.08i)T+43iT2 |
| 47 | 1−5.08T+47T2 |
| 53 | 1+(2.93−2.93i)T−53iT2 |
| 59 | 1+(−0.594+0.594i)T−59iT2 |
| 61 | 1+(2.29−5.54i)T+(−43.1−43.1i)T2 |
| 67 | 1+4.10iT−67T2 |
| 71 | 1+(9.66−4.00i)T+(50.2−50.2i)T2 |
| 73 | 1+(−0.549+0.227i)T+(51.6−51.6i)T2 |
| 79 | 1+(6.77+2.80i)T+(55.8+55.8i)T2 |
| 83 | 1+(3.35−3.35i)T−83iT2 |
| 89 | 1−14.1iT−89T2 |
| 97 | 1+(−2.37+0.982i)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
−11.50472324563020133454219322416, −10.23541767295141294448232626542, −9.701609641826944590776551376437, −8.110617165643095080608253341945, −7.85986409663544261330567908948, −7.00855601327474220909013407317, −5.97275331860395561108047625405, −5.73462917992855241908139912850, −3.22360269701160123005405295751, −1.18068411114736070104800835568,
0.15931479322803558033775783271, 2.64705570336891495145182619598, 3.49312697123903755317043223819, 4.97702338311747777738277336377, 5.82292520879262857182781697754, 7.42407121760542920297032496942, 8.767577065406845666615150019741, 9.398863745805661398044020829058, 10.17217820024102521428100719683, 10.48497940357391262011604209525