L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 − 1.5i)3-s + (−1.5 − 2.59i)4-s + 1.73i·6-s + (6.06 − 3.5i)7-s + 7i·8-s + (−1.5 + 2.59i)9-s + (5.5 + 9.52i)11-s + (−2.59 + 4.5i)12-s + 6.92·13-s − 7·14-s + (−2.5 + 4.33i)16-s + (12.1 + 21i)17-s + (2.59 − 1.5i)18-s + (3 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.433 − 0.250i)2-s + (−0.288 − 0.5i)3-s + (−0.375 − 0.649i)4-s + 0.288i·6-s + (0.866 − 0.5i)7-s + 0.875i·8-s + (−0.166 + 0.288i)9-s + (0.5 + 0.866i)11-s + (−0.216 + 0.375i)12-s + 0.532·13-s − 0.5·14-s + (−0.156 + 0.270i)16-s + (0.713 + 1.23i)17-s + (0.144 − 0.0833i)18-s + (0.157 + 0.0911i)19-s + ⋯ |
Λ(s)=(=(525s/2ΓC(s)L(s)(0.830+0.556i)Λ(3−s)
Λ(s)=(=(525s/2ΓC(s+1)L(s)(0.830+0.556i)Λ(1−s)
Degree: |
2 |
Conductor: |
525
= 3⋅52⋅7
|
Sign: |
0.830+0.556i
|
Analytic conductor: |
14.3052 |
Root analytic conductor: |
3.78222 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ525(124,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 525, ( :1), 0.830+0.556i)
|
Particular Values
L(23) |
≈ |
1.295019184 |
L(21) |
≈ |
1.295019184 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.866+1.5i)T |
| 5 | 1 |
| 7 | 1+(−6.06+3.5i)T |
good | 2 | 1+(0.866+0.5i)T+(2+3.46i)T2 |
| 11 | 1+(−5.5−9.52i)T+(−60.5+104.i)T2 |
| 13 | 1−6.92T+169T2 |
| 17 | 1+(−12.1−21i)T+(−144.5+250.i)T2 |
| 19 | 1+(−3−1.73i)T+(180.5+312.i)T2 |
| 23 | 1+(−24.2−14i)T+(264.5+458.i)T2 |
| 29 | 1+25T+841T2 |
| 31 | 1+(28.5−16.4i)T+(480.5−832.i)T2 |
| 37 | 1+(−50.2−29i)T+(684.5+1.18e3i)T2 |
| 41 | 1−3.46iT−1.68e3T2 |
| 43 | 1+26iT−1.84e3T2 |
| 47 | 1+(−38.1+66i)T+(−1.10e3−1.91e3i)T2 |
| 53 | 1+(26.8−15.5i)T+(1.40e3−2.43e3i)T2 |
| 59 | 1+(−7.5+4.33i)T+(1.74e3−3.01e3i)T2 |
| 61 | 1+(−12−6.92i)T+(1.86e3+3.22e3i)T2 |
| 67 | 1+(45.0−26i)T+(2.24e3−3.88e3i)T2 |
| 71 | 1−64T+5.04e3T2 |
| 73 | 1+(3.46+6i)T+(−2.66e3+4.61e3i)T2 |
| 79 | 1+(−8.5+14.7i)T+(−3.12e3−5.40e3i)T2 |
| 83 | 1−53.6T+6.88e3T2 |
| 89 | 1+(−69−39.8i)T+(3.96e3+6.85e3i)T2 |
| 97 | 1+91.7T+9.40e3T2 |
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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
−10.66503145146611608335478983259, −9.762018392291203785829019718573, −8.822041825759851598073770750540, −7.939944296488089614495143001092, −7.04552313116727225625616397176, −5.85675285997115856241118510316, −5.02131484312083713119126845977, −3.87216670849160274618762712321, −1.84188929529795569182209528238, −1.10923256880395524757159773713,
0.820610770197438125353071286406, 2.93057573770476948011170468968, 4.03484267833134658658519415338, 5.08184219194801493198382769646, 6.06969052970076592156088463462, 7.35131458548544408189977750382, 8.125317032918107460664325672595, 9.143094299253493697634489122135, 9.392576777121354877380601098257, 10.93868356030360667189605906237