L(s) = 1 | + (0.5 − 0.866i)5-s + (−1 − 1.73i)7-s + (−1 + 1.73i)13-s + 3·17-s + 5·19-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + (−3 − 5.19i)29-s + (−2.5 + 4.33i)31-s − 1.99·35-s + 2·37-s + (6 − 10.3i)41-s + (−4 − 6.92i)43-s + (−6 − 10.3i)47-s + (1.50 − 2.59i)49-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.277 + 0.480i)13-s + 0.727·17-s + 1.14·19-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + (−0.557 − 0.964i)29-s + (−0.449 + 0.777i)31-s − 0.338·35-s + 0.328·37-s + (0.937 − 1.62i)41-s + (−0.609 − 1.05i)43-s + (−0.875 − 1.51i)47-s + (0.214 − 0.371i)49-s + ⋯ |
Λ(s)=(=(1620s/2ΓC(s)L(s)(0.173+0.984i)Λ(2−s)
Λ(s)=(=(1620s/2ΓC(s+1/2)L(s)(0.173+0.984i)Λ(1−s)
Degree: |
2 |
Conductor: |
1620
= 22⋅34⋅5
|
Sign: |
0.173+0.984i
|
Analytic conductor: |
12.9357 |
Root analytic conductor: |
3.59663 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1620(1081,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1620, ( :1/2), 0.173+0.984i)
|
Particular Values
L(1) |
≈ |
1.524374499 |
L(21) |
≈ |
1.524374499 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+(−0.5+0.866i)T |
good | 7 | 1+(1+1.73i)T+(−3.5+6.06i)T2 |
| 11 | 1+(−5.5+9.52i)T2 |
| 13 | 1+(1−1.73i)T+(−6.5−11.2i)T2 |
| 17 | 1−3T+17T2 |
| 19 | 1−5T+19T2 |
| 23 | 1+(−1.5+2.59i)T+(−11.5−19.9i)T2 |
| 29 | 1+(3+5.19i)T+(−14.5+25.1i)T2 |
| 31 | 1+(2.5−4.33i)T+(−15.5−26.8i)T2 |
| 37 | 1−2T+37T2 |
| 41 | 1+(−6+10.3i)T+(−20.5−35.5i)T2 |
| 43 | 1+(4+6.92i)T+(−21.5+37.2i)T2 |
| 47 | 1+(6+10.3i)T+(−23.5+40.7i)T2 |
| 53 | 1−3T+53T2 |
| 59 | 1+(−3+5.19i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 |
| 67 | 1+(1−1.73i)T+(−33.5−58.0i)T2 |
| 71 | 1+12T+71T2 |
| 73 | 1+16T+73T2 |
| 79 | 1+(−0.5−0.866i)T+(−39.5+68.4i)T2 |
| 83 | 1+(7.5+12.9i)T+(−41.5+71.8i)T2 |
| 89 | 1−12T+89T2 |
| 97 | 1+(−8−13.8i)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
−9.229878683424296962663505641322, −8.540608620559694123648541416883, −7.42353419162335916711038543214, −7.02402798258473775936452474587, −5.86131873523810069205181729905, −5.15833923064090687449860109703, −4.11685894741917049189733146477, −3.28028393323370994930025626178, −1.96884560660324572121077180159, −0.63043147536621173449072760429,
1.34170110184150463463131333010, 2.78976220853864652860273553189, 3.31939411859317928661593182603, 4.70216987705528096228663152106, 5.64784628382244551014933536032, 6.15296500148472383129224226925, 7.34593049514981749829029524624, 7.79115672846270028165862531331, 8.942810603845708943826866275967, 9.627096946625867461752248566673