L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.987 − 1.42i)3-s + (0.499 − 0.866i)4-s + (2.53 + 1.46i)5-s + (1.56 + 0.738i)6-s + (−1.90 + 1.10i)7-s + 0.999i·8-s + (−1.05 + 2.81i)9-s − 2.93·10-s + (4.47 − 2.58i)11-s + (−1.72 + 0.143i)12-s + (2.72 − 2.36i)13-s + (1.10 − 1.90i)14-s + (−0.420 − 5.06i)15-s + (−0.5 − 0.866i)16-s + 2.31·17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.569 − 0.821i)3-s + (0.249 − 0.433i)4-s + (1.13 + 0.655i)5-s + (0.639 + 0.301i)6-s + (−0.721 + 0.416i)7-s + 0.353i·8-s + (−0.350 + 0.936i)9-s − 0.927·10-s + (1.34 − 0.779i)11-s + (−0.498 + 0.0414i)12-s + (0.756 − 0.654i)13-s + (0.294 − 0.510i)14-s + (−0.108 − 1.30i)15-s + (−0.125 − 0.216i)16-s + 0.560·17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.999−0.00672i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.999−0.00672i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.999−0.00672i
|
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), 0.999−0.00672i)
|
Particular Values
L(1) |
≈ |
0.939801+0.00316114i |
L(21) |
≈ |
0.939801+0.00316114i |
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+(0.987+1.42i)T |
| 13 | 1+(−2.72+2.36i)T |
good | 5 | 1+(−2.53−1.46i)T+(2.5+4.33i)T2 |
| 7 | 1+(1.90−1.10i)T+(3.5−6.06i)T2 |
| 11 | 1+(−4.47+2.58i)T+(5.5−9.52i)T2 |
| 17 | 1−2.31T+17T2 |
| 19 | 1−5.16iT−19T2 |
| 23 | 1+(−4.19+7.26i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−4.72−8.18i)T+(−14.5+25.1i)T2 |
| 31 | 1+(5.38+3.11i)T+(15.5+26.8i)T2 |
| 37 | 1+0.646iT−37T2 |
| 41 | 1+(0.674+0.389i)T+(20.5+35.5i)T2 |
| 43 | 1+(1.74+3.02i)T+(−21.5+37.2i)T2 |
| 47 | 1+(4.79−2.76i)T+(23.5−40.7i)T2 |
| 53 | 1+8.68T+53T2 |
| 59 | 1+(−2.59−1.49i)T+(29.5+51.0i)T2 |
| 61 | 1+(−0.432−0.748i)T+(−30.5+52.8i)T2 |
| 67 | 1+(9.68+5.58i)T+(33.5+58.0i)T2 |
| 71 | 1−12.9iT−71T2 |
| 73 | 1+4.27iT−73T2 |
| 79 | 1+(−1.52−2.63i)T+(−39.5+68.4i)T2 |
| 83 | 1+(3.19−1.84i)T+(41.5−71.8i)T2 |
| 89 | 1+3.34iT−89T2 |
| 97 | 1+(1.29−0.745i)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
−12.21969206416393879183587435073, −10.98276120954054199027423224860, −10.33686954685385643246276526107, −9.198824386212028918206691442785, −8.273460906700014648020247648246, −6.79632784614119182486712130100, −6.25749038393394653816838277435, −5.60068826747182151902636549193, −3.05141189846872143604909323886, −1.38589631668939481940868051337,
1.37294533765844972084907933263, 3.51112082262510937953662554516, 4.76006656287523172108565448603, 6.11686319727270000994111088148, 6.94242723384915894869528261546, 8.882541155078824731682631848701, 9.513585468672231774765538462400, 9.882854880427594147433227460446, 11.15748952145781708496375855521, 11.90359761565341945855652930097