L(s) = 1 | + (−2.56 + 1.18i)2-s − 3i·3-s + (5.20 − 6.07i)4-s − 8.43·5-s + (3.54 + 7.70i)6-s + 8.36i·7-s + (−6.20 + 21.7i)8-s − 9·9-s + (21.6 − 9.97i)10-s + 57.7·11-s + (−18.2 − 15.6i)12-s + (−18.9 − 42.8i)13-s + (−9.88 − 21.5i)14-s + 25.3i·15-s + (−9.78 − 63.2i)16-s − 72.1·17-s + ⋯ |
L(s) = 1 | + (−0.908 + 0.417i)2-s − 0.577i·3-s + (0.650 − 0.759i)4-s − 0.754·5-s + (0.241 + 0.524i)6-s + 0.451i·7-s + (−0.274 + 0.961i)8-s − 0.333·9-s + (0.685 − 0.315i)10-s + 1.58·11-s + (−0.438 − 0.375i)12-s + (−0.404 − 0.914i)13-s + (−0.188 − 0.410i)14-s + 0.435i·15-s + (−0.152 − 0.988i)16-s − 1.03·17-s + ⋯ |
Λ(s)=(=(312s/2ΓC(s)L(s)(−0.138−0.990i)Λ(4−s)
Λ(s)=(=(312s/2ΓC(s+3/2)L(s)(−0.138−0.990i)Λ(1−s)
Degree: |
2 |
Conductor: |
312
= 23⋅3⋅13
|
Sign: |
−0.138−0.990i
|
Analytic conductor: |
18.4085 |
Root analytic conductor: |
4.29052 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ312(181,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 312, ( :3/2), −0.138−0.990i)
|
Particular Values
L(2) |
≈ |
0.5696656030 |
L(21) |
≈ |
0.5696656030 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(2.56−1.18i)T |
| 3 | 1+3iT |
| 13 | 1+(18.9+42.8i)T |
good | 5 | 1+8.43T+125T2 |
| 7 | 1−8.36iT−343T2 |
| 11 | 1−57.7T+1.33e3T2 |
| 17 | 1+72.1T+4.91e3T2 |
| 19 | 1+144.T+6.85e3T2 |
| 23 | 1−168.T+1.21e4T2 |
| 29 | 1−96.0iT−2.43e4T2 |
| 31 | 1−59.3iT−2.97e4T2 |
| 37 | 1−187.T+5.06e4T2 |
| 41 | 1−211.iT−6.89e4T2 |
| 43 | 1−160.iT−7.95e4T2 |
| 47 | 1−539.iT−1.03e5T2 |
| 53 | 1−583.iT−1.48e5T2 |
| 59 | 1−236.T+2.05e5T2 |
| 61 | 1−438.iT−2.26e5T2 |
| 67 | 1+639.T+3.00e5T2 |
| 71 | 1−79.3iT−3.57e5T2 |
| 73 | 1+40.6iT−3.89e5T2 |
| 79 | 1−807.T+4.93e5T2 |
| 83 | 1+1.39e3T+5.71e5T2 |
| 89 | 1+1.05e3iT−7.04e5T2 |
| 97 | 1−1.00e3iT−9.12e5T2 |
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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
−11.36758704023005427298164186239, −10.67333580104224797058617050729, −9.209198080504321470766478050528, −8.703235072671818573041468798963, −7.70387068666513273213906284704, −6.78123080884047833914807031014, −6.03038776181670562336465039496, −4.47303947559151765434326158569, −2.68780812381951862336972370561, −1.17155600331547277483010712389,
0.31673999890319670477211614783, 2.08510050101995510050238968327, 3.82392882301930319754646899818, 4.31893223469430979096144732946, 6.51337912536819635998045131266, 7.13860678705783575652665710941, 8.534954042337560323892284949792, 9.046205460334416943457082382762, 10.00787951198613910810703880247, 11.13145159046280639287414893815