L(s) = 1 | + (3.46 − 2i)2-s + (9.95 + 11.9i)3-s + (7.99 − 13.8i)4-s + (41.5 + 72.0i)5-s + (58.4 + 21.6i)6-s + (−128. − 15.2i)7-s − 63.9i·8-s + (−44.9 + 238. i)9-s + (288. + 166. i)10-s + (455. + 263. i)11-s + (245. − 41.9i)12-s − 776. i·13-s + (−476. + 204. i)14-s + (−450. + 1.21e3i)15-s + (−128 − 221. i)16-s + (236. − 409. i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.638 + 0.769i)3-s + (0.249 − 0.433i)4-s + (0.744 + 1.28i)5-s + (0.663 + 0.245i)6-s + (−0.993 − 0.117i)7-s − 0.353i·8-s + (−0.184 + 0.982i)9-s + (0.911 + 0.526i)10-s + (1.13 + 0.655i)11-s + (0.492 − 0.0839i)12-s − 1.27i·13-s + (−0.649 + 0.279i)14-s + (−0.516 + 1.39i)15-s + (−0.125 − 0.216i)16-s + (0.198 − 0.343i)17-s + ⋯ |
Λ(s)=(=(42s/2ΓC(s)L(s)(0.763−0.645i)Λ(6−s)
Λ(s)=(=(42s/2ΓC(s+5/2)L(s)(0.763−0.645i)Λ(1−s)
Degree: |
2 |
Conductor: |
42
= 2⋅3⋅7
|
Sign: |
0.763−0.645i
|
Analytic conductor: |
6.73612 |
Root analytic conductor: |
2.59540 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ42(17,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 42, ( :5/2), 0.763−0.645i)
|
Particular Values
L(3) |
≈ |
2.63582+0.964384i |
L(21) |
≈ |
2.63582+0.964384i |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−3.46+2i)T |
| 3 | 1+(−9.95−11.9i)T |
| 7 | 1+(128.+15.2i)T |
good | 5 | 1+(−41.5−72.0i)T+(−1.56e3+2.70e3i)T2 |
| 11 | 1+(−455.−263.i)T+(8.05e4+1.39e5i)T2 |
| 13 | 1+776.iT−3.71e5T2 |
| 17 | 1+(−236.+409.i)T+(−7.09e5−1.22e6i)T2 |
| 19 | 1+(−1.71e3+990.i)T+(1.23e6−2.14e6i)T2 |
| 23 | 1+(1.49e3−863.i)T+(3.21e6−5.57e6i)T2 |
| 29 | 1+6.04e3iT−2.05e7T2 |
| 31 | 1+(7.50e3+4.33e3i)T+(1.43e7+2.47e7i)T2 |
| 37 | 1+(−1.50e3−2.61e3i)T+(−3.46e7+6.00e7i)T2 |
| 41 | 1+1.63e3T+1.15e8T2 |
| 43 | 1+1.19e3T+1.47e8T2 |
| 47 | 1+(−1.93e3−3.35e3i)T+(−1.14e8+1.98e8i)T2 |
| 53 | 1+(−3.24e3−1.87e3i)T+(2.09e8+3.62e8i)T2 |
| 59 | 1+(−3.24e3+5.61e3i)T+(−3.57e8−6.19e8i)T2 |
| 61 | 1+(−3.85e4+2.22e4i)T+(4.22e8−7.31e8i)T2 |
| 67 | 1+(1.21e4−2.11e4i)T+(−6.75e8−1.16e9i)T2 |
| 71 | 1−8.35e4iT−1.80e9T2 |
| 73 | 1+(−1.11e4−6.41e3i)T+(1.03e9+1.79e9i)T2 |
| 79 | 1+(2.01e4+3.49e4i)T+(−1.53e9+2.66e9i)T2 |
| 83 | 1+7.28e4T+3.93e9T2 |
| 89 | 1+(2.92e4+5.07e4i)T+(−2.79e9+4.83e9i)T2 |
| 97 | 1+1.01e3iT−8.58e9T2 |
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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
−14.92222019511902153311385666716, −14.06551645243701500054580938720, −13.12682863665083909708274394662, −11.39397248150130743083170646151, −10.04088135940735687188022390389, −9.616589036066281380033118348442, −7.21530191020795344088149084624, −5.76221557846472752893220133722, −3.69473379189226860246875555721, −2.60152868956206793374342082424,
1.51120121513796403814305100814, 3.67600093243620725543663626927, 5.74951920659998844506178953976, 6.87553710736478587349155230822, 8.717680605751621267961412066836, 9.402779986374174262024610751867, 11.98552130732788742357072364665, 12.70226427291572458849210643155, 13.75239060250356885112777821269, 14.40141018863389372915508890586