L(s) = 1 | + (−5.65 + 5.65i)2-s + (46.6 − 3.26i)3-s − 64.0i·4-s + (−245. + 282. i)6-s + (918. + 918. i)7-s + (362. + 362. i)8-s + (2.16e3 − 304. i)9-s − 7.13e3i·11-s + (−209. − 2.98e3i)12-s + (1.04e4 − 1.04e4i)13-s − 1.03e4·14-s − 4.09e3·16-s + (−1.98e4 + 1.98e4i)17-s + (−1.05e4 + 1.39e4i)18-s − 4.24e4i·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.997 − 0.0698i)3-s − 0.500i·4-s + (−0.463 + 0.533i)6-s + (1.01 + 1.01i)7-s + (0.250 + 0.250i)8-s + (0.990 − 0.139i)9-s − 1.61i·11-s + (−0.0349 − 0.498i)12-s + (1.31 − 1.31i)13-s − 1.01·14-s − 0.250·16-s + (−0.980 + 0.980i)17-s + (−0.425 + 0.564i)18-s − 1.41i·19-s + ⋯ |
Λ(s)=(=(150s/2ΓC(s)L(s)(0.965+0.259i)Λ(8−s)
Λ(s)=(=(150s/2ΓC(s+7/2)L(s)(0.965+0.259i)Λ(1−s)
Degree: |
2 |
Conductor: |
150
= 2⋅3⋅52
|
Sign: |
0.965+0.259i
|
Analytic conductor: |
46.8577 |
Root analytic conductor: |
6.84527 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ150(107,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 150, ( :7/2), 0.965+0.259i)
|
Particular Values
L(4) |
≈ |
2.726520127 |
L(21) |
≈ |
2.726520127 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(5.65−5.65i)T |
| 3 | 1+(−46.6+3.26i)T |
| 5 | 1 |
good | 7 | 1+(−918.−918.i)T+8.23e5iT2 |
| 11 | 1+7.13e3iT−1.94e7T2 |
| 13 | 1+(−1.04e4+1.04e4i)T−6.27e7iT2 |
| 17 | 1+(1.98e4−1.98e4i)T−4.10e8iT2 |
| 19 | 1+4.24e4iT−8.93e8T2 |
| 23 | 1+(−4.59e3−4.59e3i)T+3.40e9iT2 |
| 29 | 1+1.04e5T+1.72e10T2 |
| 31 | 1+3.87e4T+2.75e10T2 |
| 37 | 1+(1.24e5+1.24e5i)T+9.49e10iT2 |
| 41 | 1−3.28e5iT−1.94e11T2 |
| 43 | 1+(−1.23e5+1.23e5i)T−2.71e11iT2 |
| 47 | 1+(−6.31e5+6.31e5i)T−5.06e11iT2 |
| 53 | 1+(7.06e5+7.06e5i)T+1.17e12iT2 |
| 59 | 1−4.01e5T+2.48e12T2 |
| 61 | 1−2.71e6T+3.14e12T2 |
| 67 | 1+(1.30e6+1.30e6i)T+6.06e12iT2 |
| 71 | 1−1.05e6iT−9.09e12T2 |
| 73 | 1+(−2.39e6+2.39e6i)T−1.10e13iT2 |
| 79 | 1+2.40e6iT−1.92e13T2 |
| 83 | 1+(−3.09e6−3.09e6i)T+2.71e13iT2 |
| 89 | 1−2.35e6T+4.42e13T2 |
| 97 | 1+(−7.53e6−7.53e6i)T+8.07e13iT2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.30104202073015554227164082958, −10.68224163035526211699234628680, −8.930024808059814535099553513752, −8.632601994762727625656300943724, −7.87733240912223576202939818848, −6.32624213009055298751949693321, −5.27817586366752554116072539014, −3.55395797136896938291135450322, −2.19676632306430107644387296615, −0.832268629222879466325752514299,
1.38303109028802929021532996670, 2.05883264885694853108446902009, 3.89407619009794801079697357880, 4.50188297721779130588643918873, 6.94465746869512391311094022825, 7.65231355701250160961608560860, 8.748482784331884450250369400841, 9.614754750311767792652973779124, 10.61043673567060272558749937414, 11.54126763057297224050749813932