L(s) = 1 | + (0.866 − 0.5i)2-s + (1.66 + 0.480i)3-s + (0.499 − 0.866i)4-s + (0.515 + 0.297i)5-s + (1.68 − 0.416i)6-s + (−1.45 + 0.838i)7-s − 0.999i·8-s + (2.53 + 1.59i)9-s + 0.594·10-s + (−0.416 + 0.240i)11-s + (1.24 − 1.20i)12-s + (−2.27 − 2.79i)13-s + (−0.838 + 1.45i)14-s + (0.714 + 0.742i)15-s + (−0.5 − 0.866i)16-s − 2.09·17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.960 + 0.277i)3-s + (0.249 − 0.433i)4-s + (0.230 + 0.132i)5-s + (0.686 − 0.169i)6-s + (−0.548 + 0.316i)7-s − 0.353i·8-s + (0.846 + 0.532i)9-s + 0.188·10-s + (−0.125 + 0.0724i)11-s + (0.360 − 0.346i)12-s + (−0.630 − 0.776i)13-s + (−0.224 + 0.388i)14-s + (0.184 + 0.191i)15-s + (−0.125 − 0.216i)16-s − 0.507·17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.973+0.227i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.973+0.227i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.973+0.227i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), 0.973+0.227i)
|
Particular Values
L(1) |
≈ |
2.13855−0.246526i |
L(21) |
≈ |
2.13855−0.246526i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.866+0.5i)T |
| 3 | 1+(−1.66−0.480i)T |
| 13 | 1+(2.27+2.79i)T |
good | 5 | 1+(−0.515−0.297i)T+(2.5+4.33i)T2 |
| 7 | 1+(1.45−0.838i)T+(3.5−6.06i)T2 |
| 11 | 1+(0.416−0.240i)T+(5.5−9.52i)T2 |
| 17 | 1+2.09T+17T2 |
| 19 | 1+0.480iT−19T2 |
| 23 | 1+(1.83−3.17i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−1.23−2.13i)T+(−14.5+25.1i)T2 |
| 31 | 1+(0.993+0.573i)T+(15.5+26.8i)T2 |
| 37 | 1+3.65iT−37T2 |
| 41 | 1+(−8.58−4.95i)T+(20.5+35.5i)T2 |
| 43 | 1+(3.45+5.98i)T+(−21.5+37.2i)T2 |
| 47 | 1+(5.40−3.12i)T+(23.5−40.7i)T2 |
| 53 | 1+5.08T+53T2 |
| 59 | 1+(−8.13−4.69i)T+(29.5+51.0i)T2 |
| 61 | 1+(3.90+6.76i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−12.4−7.19i)T+(33.5+58.0i)T2 |
| 71 | 1−6.51iT−71T2 |
| 73 | 1+5.91iT−73T2 |
| 79 | 1+(−1.02−1.78i)T+(−39.5+68.4i)T2 |
| 83 | 1+(9.57−5.53i)T+(41.5−71.8i)T2 |
| 89 | 1+9.48iT−89T2 |
| 97 | 1+(−8.41+4.85i)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
−12.48235566612562226359361744002, −11.14959010777295727765195358777, −10.06733077209014353526528237174, −9.501276667028101183915521040867, −8.261227116370834689915837650831, −7.12834929686288526865171161808, −5.82102592374527322777891613318, −4.54936170535268934751209010602, −3.27195241860241752466794980364, −2.26076031267130636164631320936,
2.20041243820958489385173564026, 3.56344703688578581479097108876, 4.69444008951545229445954931308, 6.29441893844356062568527921009, 7.12032331964608550877654080367, 8.138948933392736250176010346745, 9.228600904119461894514039439408, 10.05540701826271799474639862294, 11.50070555154639086049904901647, 12.60127086902683813831774516150