L(s) = 1 | + (−0.5 − 1.53i)3-s + (−2.45 − 1.78i)5-s + (0.309 − 0.951i)7-s + (0.309 − 0.224i)9-s + (−3.31 + 0.141i)11-s + (−1.54 + 1.11i)13-s + (−1.51 + 4.66i)15-s + (−3.90 − 2.83i)17-s + (2.08 + 6.40i)19-s − 1.61·21-s + 7.99·23-s + (1.29 + 4.00i)25-s + (−4.42 − 3.21i)27-s + (0.254 − 0.782i)29-s + (−7.96 + 5.78i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.888i)3-s + (−1.09 − 0.797i)5-s + (0.116 − 0.359i)7-s + (0.103 − 0.0748i)9-s + (−0.999 + 0.0425i)11-s + (−0.427 + 0.310i)13-s + (−0.391 + 1.20i)15-s + (−0.947 − 0.688i)17-s + (0.477 + 1.46i)19-s − 0.353·21-s + 1.66·23-s + (0.259 + 0.800i)25-s + (−0.851 − 0.619i)27-s + (0.0472 − 0.145i)29-s + (−1.43 + 1.03i)31-s + ⋯ |
Λ(s)=(=(616s/2ΓC(s)L(s)(−0.819−0.572i)Λ(2−s)
Λ(s)=(=(616s/2ΓC(s+1/2)L(s)(−0.819−0.572i)Λ(1−s)
Degree: |
2 |
Conductor: |
616
= 23⋅7⋅11
|
Sign: |
−0.819−0.572i
|
Analytic conductor: |
4.91878 |
Root analytic conductor: |
2.21783 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ616(113,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 616, ( :1/2), −0.819−0.572i)
|
Particular Values
L(1) |
≈ |
0.105011+0.333823i |
L(21) |
≈ |
0.105011+0.333823i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1+(−0.309+0.951i)T |
| 11 | 1+(3.31−0.141i)T |
good | 3 | 1+(0.5+1.53i)T+(−2.42+1.76i)T2 |
| 5 | 1+(2.45+1.78i)T+(1.54+4.75i)T2 |
| 13 | 1+(1.54−1.11i)T+(4.01−12.3i)T2 |
| 17 | 1+(3.90+2.83i)T+(5.25+16.1i)T2 |
| 19 | 1+(−2.08−6.40i)T+(−15.3+11.1i)T2 |
| 23 | 1−7.99T+23T2 |
| 29 | 1+(−0.254+0.782i)T+(−23.4−17.0i)T2 |
| 31 | 1+(7.96−5.78i)T+(9.57−29.4i)T2 |
| 37 | 1+(1.70−5.25i)T+(−29.9−21.7i)T2 |
| 41 | 1+(2.83+8.72i)T+(−33.1+24.0i)T2 |
| 43 | 1+11.1T+43T2 |
| 47 | 1+(−1.79−5.51i)T+(−38.0+27.6i)T2 |
| 53 | 1+(−5.95+4.32i)T+(16.3−50.4i)T2 |
| 59 | 1+(0.565−1.74i)T+(−47.7−34.6i)T2 |
| 61 | 1+(7.77+5.64i)T+(18.8+58.0i)T2 |
| 67 | 1−2.23T+67T2 |
| 71 | 1+(−3.31−2.40i)T+(21.9+67.5i)T2 |
| 73 | 1+(−3.78+11.6i)T+(−59.0−42.9i)T2 |
| 79 | 1+(0.0773−0.0562i)T+(24.4−75.1i)T2 |
| 83 | 1+(11.1+8.08i)T+(25.6+78.9i)T2 |
| 89 | 1+12.6T+89T2 |
| 97 | 1+(−11.8+8.63i)T+(29.9−92.2i)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
−10.20197644753609844102452471360, −9.046373855212558988765623560517, −8.173631150258138887922878972315, −7.36559436662237646304753240294, −6.87187680137816115014622539580, −5.36853130571650135624872196246, −4.58773279446453017572606664801, −3.37094004506957232896216644328, −1.62597250112864602654819761538, −0.19810957655693543483133562654,
2.57474308921371351057231808933, 3.60799160708345667537759345483, 4.70565197717388844269929775653, 5.38348133446850940149213814908, 6.92043106421553605727541914203, 7.49435715947428966604313662013, 8.570140993762497584395396906895, 9.505676638317444792803837981019, 10.55065080998886066051409805927, 11.06461446968718422887470221001