L(s) = 1 | + (−0.645 + 1.98i)3-s + (1.70 − 1.24i)5-s + (−0.309 − 0.951i)7-s + (−1.10 − 0.801i)9-s + (−2.67 + 1.96i)11-s + (4.98 + 3.62i)13-s + (1.36 + 4.19i)15-s + (2.23 − 1.62i)17-s + (−0.142 + 0.438i)19-s + 2.08·21-s + 6.31·23-s + (−0.165 + 0.508i)25-s + (−2.76 + 2.00i)27-s + (0.693 + 2.13i)29-s + (−2.67 − 1.94i)31-s + ⋯ |
L(s) = 1 | + (−0.372 + 1.14i)3-s + (0.764 − 0.555i)5-s + (−0.116 − 0.359i)7-s + (−0.367 − 0.267i)9-s + (−0.805 + 0.592i)11-s + (1.38 + 1.00i)13-s + (0.352 + 1.08i)15-s + (0.541 − 0.393i)17-s + (−0.0326 + 0.100i)19-s + 0.455·21-s + 1.31·23-s + (−0.0330 + 0.101i)25-s + (−0.532 + 0.386i)27-s + (0.128 + 0.396i)29-s + (−0.480 − 0.349i)31-s + ⋯ |
Λ(s)=(=(616s/2ΓC(s)L(s)(0.281−0.959i)Λ(2−s)
Λ(s)=(=(616s/2ΓC(s+1/2)L(s)(0.281−0.959i)Λ(1−s)
Degree: |
2 |
Conductor: |
616
= 23⋅7⋅11
|
Sign: |
0.281−0.959i
|
Analytic conductor: |
4.91878 |
Root analytic conductor: |
2.21783 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ616(169,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 616, ( :1/2), 0.281−0.959i)
|
Particular Values
L(1) |
≈ |
1.19364+0.894068i |
L(21) |
≈ |
1.19364+0.894068i |
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+(2.67−1.96i)T |
good | 3 | 1+(0.645−1.98i)T+(−2.42−1.76i)T2 |
| 5 | 1+(−1.70+1.24i)T+(1.54−4.75i)T2 |
| 13 | 1+(−4.98−3.62i)T+(4.01+12.3i)T2 |
| 17 | 1+(−2.23+1.62i)T+(5.25−16.1i)T2 |
| 19 | 1+(0.142−0.438i)T+(−15.3−11.1i)T2 |
| 23 | 1−6.31T+23T2 |
| 29 | 1+(−0.693−2.13i)T+(−23.4+17.0i)T2 |
| 31 | 1+(2.67+1.94i)T+(9.57+29.4i)T2 |
| 37 | 1+(−2.26−6.95i)T+(−29.9+21.7i)T2 |
| 41 | 1+(−0.0315+0.0970i)T+(−33.1−24.0i)T2 |
| 43 | 1−3.33T+43T2 |
| 47 | 1+(2.56−7.88i)T+(−38.0−27.6i)T2 |
| 53 | 1+(10.1+7.37i)T+(16.3+50.4i)T2 |
| 59 | 1+(−0.636−1.95i)T+(−47.7+34.6i)T2 |
| 61 | 1+(2.15−1.56i)T+(18.8−58.0i)T2 |
| 67 | 1−5.10T+67T2 |
| 71 | 1+(−11.4+8.31i)T+(21.9−67.5i)T2 |
| 73 | 1+(0.651+2.00i)T+(−59.0+42.9i)T2 |
| 79 | 1+(−1.93−1.40i)T+(24.4+75.1i)T2 |
| 83 | 1+(−6.49+4.72i)T+(25.6−78.9i)T2 |
| 89 | 1+10.9T+89T2 |
| 97 | 1+(9.99+7.26i)T+(29.9+92.2i)T2 |
show more | |
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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
−10.87409295717971019822567427717, −9.692302646732514764244419132142, −9.505418521297275640777555345690, −8.412982415576980743644359736857, −7.18228643219678665725729532878, −6.05789220271712297991806820819, −5.11924523453322672450319476154, −4.47783737305801165923088126964, −3.29852009936202648505077446282, −1.53160812047034509785150573316,
0.979011188043482530506362601443, 2.37551736154765562121957564007, 3.45602401570214832719401438291, 5.42566359333246566965838394674, 5.97111487880539918218884753873, 6.71064208366805988982927567297, 7.75937103700763838288745068622, 8.495086459075226834662463790238, 9.644783790392192331333720186226, 10.73345042976541322331182293827