L(s) = 1 | + (1 + i)2-s + 2i·4-s + 3.73·5-s + (−3 − 1.73i)7-s + (−2 + 2i)8-s + (3.73 + 3.73i)10-s + (1 + 1.73i)11-s + (2.59 + 2.5i)13-s + (−1.26 − 4.73i)14-s − 4·16-s + (−0.232 + 0.401i)17-s + (−0.633 + 1.09i)19-s + 7.46i·20-s + (−0.732 + 2.73i)22-s + (4.09 + 7.09i)23-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + i·4-s + 1.66·5-s + (−1.13 − 0.654i)7-s + (−0.707 + 0.707i)8-s + (1.18 + 1.18i)10-s + (0.301 + 0.522i)11-s + (0.720 + 0.693i)13-s + (−0.338 − 1.26i)14-s − 16-s + (−0.0562 + 0.0974i)17-s + (−0.145 + 0.251i)19-s + 1.66i·20-s + (−0.156 + 0.582i)22-s + (0.854 + 1.48i)23-s + ⋯ |
Λ(s)=(=(936s/2ΓC(s)L(s)(−0.00641−0.999i)Λ(2−s)
Λ(s)=(=(936s/2ΓC(s+1/2)L(s)(−0.00641−0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
936
= 23⋅32⋅13
|
Sign: |
−0.00641−0.999i
|
Analytic conductor: |
7.47399 |
Root analytic conductor: |
2.73386 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ936(829,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 936, ( :1/2), −0.00641−0.999i)
|
Particular Values
L(1) |
≈ |
1.92555+1.93793i |
L(21) |
≈ |
1.92555+1.93793i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1−i)T |
| 3 | 1 |
| 13 | 1+(−2.59−2.5i)T |
good | 5 | 1−3.73T+5T2 |
| 7 | 1+(3+1.73i)T+(3.5+6.06i)T2 |
| 11 | 1+(−1−1.73i)T+(−5.5+9.52i)T2 |
| 17 | 1+(0.232−0.401i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.633−1.09i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−4.09−7.09i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−2.59+1.5i)T+(14.5−25.1i)T2 |
| 31 | 1−4.73iT−31T2 |
| 37 | 1+(2.13+3.69i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−7.96+4.59i)T+(20.5−35.5i)T2 |
| 43 | 1+(2.19+1.26i)T+(21.5+37.2i)T2 |
| 47 | 1+6.73iT−47T2 |
| 53 | 1+3.92iT−53T2 |
| 59 | 1+(−0.267+0.464i)T+(−29.5−51.0i)T2 |
| 61 | 1+(0.866+0.5i)T+(30.5+52.8i)T2 |
| 67 | 1+(3.63+6.29i)T+(−33.5+58.0i)T2 |
| 71 | 1+(8.02+4.63i)T+(35.5+61.4i)T2 |
| 73 | 1+1.73iT−73T2 |
| 79 | 1+10.3T+79T2 |
| 83 | 1−1.46T+83T2 |
| 89 | 1+(6.46−3.73i)T+(44.5−77.0i)T2 |
| 97 | 1+(5.19+3i)T+(48.5+84.0i)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
−10.08412023539536633530782241543, −9.360233796138112197905075528220, −8.772481965723135143715989135779, −7.28157733645469042209151360310, −6.70173035781153376575508480834, −6.03821756743108257369137222180, −5.25660295682820022210260917027, −4.05433764311912467631680730520, −3.10345940500873424179208467540, −1.76012333728908991450330955031,
1.11551106814068224236087518847, 2.59276568702195517920161391102, 3.03812214052929983463667137519, 4.53837728811293886892487644309, 5.70156955878799040095552884800, 6.08952262550183486745177740393, 6.72431351913033562291945935019, 8.667509117010840077712667965580, 9.231200847925196360671801306046, 9.975482804880099460005594972141