L(s) = 1 | + (1.80 − 1.04i)2-s + (0.713 − 2.66i)3-s + (1.17 − 2.04i)4-s + (−2.22 − 0.194i)5-s + (−1.48 − 5.55i)6-s + (1.45 − 2.52i)7-s − 0.750i·8-s + (−3.97 − 2.29i)9-s + (−4.23 + 1.97i)10-s + (−0.00681 + 0.0254i)11-s + (−4.59 − 4.59i)12-s − 6.07i·14-s + (−2.10 + 5.79i)15-s + (1.57 + 2.72i)16-s + (2.76 − 0.741i)17-s − 9.58·18-s + ⋯ |
L(s) = 1 | + (1.27 − 0.738i)2-s + (0.411 − 1.53i)3-s + (0.589 − 1.02i)4-s + (−0.996 − 0.0869i)5-s + (−0.607 − 2.26i)6-s + (0.550 − 0.952i)7-s − 0.265i·8-s + (−1.32 − 0.765i)9-s + (−1.33 + 0.624i)10-s + (−0.00205 + 0.00767i)11-s + (−1.32 − 1.32i)12-s − 1.62i·14-s + (−0.543 + 1.49i)15-s + (0.393 + 0.682i)16-s + (0.671 − 0.179i)17-s − 2.26·18-s + ⋯ |
Λ(s)=(=(845s/2ΓC(s)L(s)(−0.995+0.0996i)Λ(2−s)
Λ(s)=(=(845s/2ΓC(s+1/2)L(s)(−0.995+0.0996i)Λ(1−s)
Degree: |
2 |
Conductor: |
845
= 5⋅132
|
Sign: |
−0.995+0.0996i
|
Analytic conductor: |
6.74735 |
Root analytic conductor: |
2.59756 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ845(418,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 845, ( :1/2), −0.995+0.0996i)
|
Particular Values
L(1) |
≈ |
0.148181−2.96806i |
L(21) |
≈ |
0.148181−2.96806i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(2.22+0.194i)T |
| 13 | 1 |
good | 2 | 1+(−1.80+1.04i)T+(1−1.73i)T2 |
| 3 | 1+(−0.713+2.66i)T+(−2.59−1.5i)T2 |
| 7 | 1+(−1.45+2.52i)T+(−3.5−6.06i)T2 |
| 11 | 1+(0.00681−0.0254i)T+(−9.52−5.5i)T2 |
| 17 | 1+(−2.76+0.741i)T+(14.7−8.5i)T2 |
| 19 | 1+(4.62−1.23i)T+(16.4−9.5i)T2 |
| 23 | 1+(−0.358−0.0961i)T+(19.9+11.5i)T2 |
| 29 | 1+(−3.62+2.09i)T+(14.5−25.1i)T2 |
| 31 | 1+(−0.835+0.835i)T−31iT2 |
| 37 | 1+(−3.22−5.58i)T+(−18.5+32.0i)T2 |
| 41 | 1+(7.57+2.02i)T+(35.5+20.5i)T2 |
| 43 | 1+(1.79+6.69i)T+(−37.2+21.5i)T2 |
| 47 | 1+0.833T+47T2 |
| 53 | 1+(−0.902−0.902i)T+53iT2 |
| 59 | 1+(0.387+1.44i)T+(−51.0+29.5i)T2 |
| 61 | 1+(−5.35+9.26i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−10.6+6.15i)T+(33.5−58.0i)T2 |
| 71 | 1+(0.957+3.57i)T+(−61.4+35.5i)T2 |
| 73 | 1−15.0iT−73T2 |
| 79 | 1−4.25iT−79T2 |
| 83 | 1+1.31T+83T2 |
| 89 | 1+(3.23+0.867i)T+(77.0+44.5i)T2 |
| 97 | 1+(−0.351−0.202i)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
−10.18470758661346946929691766282, −8.395876665793104221387523960098, −8.088965221148522626493212234379, −7.14340885764464465147773975052, −6.38678083324100355674536080162, −5.07108083174844059704455061735, −4.13445575787222675665757773538, −3.28433751494848353387665093427, −2.13622824670605872763107958002, −0.977426736453809618221803231864,
2.79416101496199824779178197840, 3.63924022647383167302217473551, 4.46419115595297191079511465865, 4.99531706514008983387559648215, 5.89258422594714363278926331891, 7.02626210382514990640881318651, 8.194563550430690899110166920646, 8.671192846114977843700829906553, 9.759841628158983727805240508019, 10.66407259664774653049685333770