L(s) = 1 | + (1.79 − 1.79i)5-s + (1.02 + 3.83i)7-s + (−1.05 + 3.95i)11-s + (1.54 + 3.25i)13-s + (−1.68 + 2.91i)17-s + (−7.63 + 2.04i)19-s + (−1.54 − 2.67i)23-s − 1.41i·25-s + (7.02 − 4.05i)29-s + (0.618 + 0.618i)31-s + (8.70 + 5.02i)35-s + (−8.92 − 2.39i)37-s + (6.25 + 1.67i)41-s + (8.40 + 4.85i)43-s + (4.37 + 4.37i)47-s + ⋯ |
L(s) = 1 | + (0.801 − 0.801i)5-s + (0.388 + 1.44i)7-s + (−0.319 + 1.19i)11-s + (0.428 + 0.903i)13-s + (−0.408 + 0.707i)17-s + (−1.75 + 0.469i)19-s + (−0.322 − 0.558i)23-s − 0.283i·25-s + (1.30 − 0.753i)29-s + (0.111 + 0.111i)31-s + (1.47 + 0.849i)35-s + (−1.46 − 0.392i)37-s + (0.977 + 0.261i)41-s + (1.28 + 0.740i)43-s + (0.637 + 0.637i)47-s + ⋯ |
Λ(s)=(=(936s/2ΓC(s)L(s)(0.385−0.922i)Λ(2−s)
Λ(s)=(=(936s/2ΓC(s+1/2)L(s)(0.385−0.922i)Λ(1−s)
Degree: |
2 |
Conductor: |
936
= 23⋅32⋅13
|
Sign: |
0.385−0.922i
|
Analytic conductor: |
7.47399 |
Root analytic conductor: |
2.73386 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ936(89,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 936, ( :1/2), 0.385−0.922i)
|
Particular Values
L(1) |
≈ |
1.39421+0.928563i |
L(21) |
≈ |
1.39421+0.928563i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 13 | 1+(−1.54−3.25i)T |
good | 5 | 1+(−1.79+1.79i)T−5iT2 |
| 7 | 1+(−1.02−3.83i)T+(−6.06+3.5i)T2 |
| 11 | 1+(1.05−3.95i)T+(−9.52−5.5i)T2 |
| 17 | 1+(1.68−2.91i)T+(−8.5−14.7i)T2 |
| 19 | 1+(7.63−2.04i)T+(16.4−9.5i)T2 |
| 23 | 1+(1.54+2.67i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−7.02+4.05i)T+(14.5−25.1i)T2 |
| 31 | 1+(−0.618−0.618i)T+31iT2 |
| 37 | 1+(8.92+2.39i)T+(32.0+18.5i)T2 |
| 41 | 1+(−6.25−1.67i)T+(35.5+20.5i)T2 |
| 43 | 1+(−8.40−4.85i)T+(21.5+37.2i)T2 |
| 47 | 1+(−4.37−4.37i)T+47iT2 |
| 53 | 1+13.8iT−53T2 |
| 59 | 1+(4.03−1.07i)T+(51.0−29.5i)T2 |
| 61 | 1+(4.06−7.04i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−3.17+11.8i)T+(−58.0−33.5i)T2 |
| 71 | 1+(−0.166−0.622i)T+(−61.4+35.5i)T2 |
| 73 | 1+(0.788−0.788i)T−73iT2 |
| 79 | 1−15.1T+79T2 |
| 83 | 1+(−0.0917+0.0917i)T−83iT2 |
| 89 | 1+(−3.46+12.9i)T+(−77.0−44.5i)T2 |
| 97 | 1+(−6.54+1.75i)T+(84.0−48.5i)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
−10.12734306109965808860143860770, −9.160216045986610850894844908401, −8.751016493209495025434203745649, −7.963519887687340279782867399256, −6.48129625830260231811587742754, −5.98190507810529795061824111786, −4.90028746799820373654680544722, −4.25740507705922591602906053996, −2.27927180634260617762385168223, −1.84942500545357187732907356296,
0.78280293658607724609184175670, 2.41787166938535934820320777784, 3.44837998344225563929652024316, 4.52597896782864703961760364029, 5.70271502511303480533420665920, 6.50632165148031500324153958638, 7.26932192364173959074260416084, 8.217785457135138524801706597320, 9.024275151044624378964900883812, 10.32299588620774394955190587074