L(s) = 1 | + (13.5 + 23.3i)3-s + (−50 + 86.6i)5-s + (−364.5 + 631. i)9-s + (−1.38e3 − 2.40e3i)11-s − 3.29e3·13-s − 2.70e3·15-s + (−2.95e3 − 5.10e3i)17-s + (−3.32e3 + 5.75e3i)19-s + (−991 + 1.71e3i)23-s + (3.40e4 + 5.89e4i)25-s − 1.96e4·27-s − 2.08e5·29-s + (5.88e4 + 1.02e5i)31-s + (3.74e4 − 6.48e4i)33-s + (1.67e5 − 2.90e5i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.178 + 0.309i)5-s + (−0.166 + 0.288i)9-s + (−0.314 − 0.544i)11-s − 0.415·13-s − 0.206·15-s + (−0.145 − 0.252i)17-s + (−0.111 + 0.192i)19-s + (−0.0169 + 0.0294i)23-s + (0.435 + 0.755i)25-s − 0.192·27-s − 1.58·29-s + (0.355 + 0.615i)31-s + (0.181 − 0.314i)33-s + (0.544 − 0.943i)37-s + ⋯ |
Λ(s)=(=(588s/2ΓC(s)L(s)(0.991+0.126i)Λ(8−s)
Λ(s)=(=(588s/2ΓC(s+7/2)L(s)(0.991+0.126i)Λ(1−s)
Degree: |
2 |
Conductor: |
588
= 22⋅3⋅72
|
Sign: |
0.991+0.126i
|
Analytic conductor: |
183.682 |
Root analytic conductor: |
13.5529 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ588(373,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 588, ( :7/2), 0.991+0.126i)
|
Particular Values
L(4) |
≈ |
1.674718660 |
L(21) |
≈ |
1.674718660 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(−13.5−23.3i)T |
| 7 | 1 |
good | 5 | 1+(50−86.6i)T+(−3.90e4−6.76e4i)T2 |
| 11 | 1+(1.38e3+2.40e3i)T+(−9.74e6+1.68e7i)T2 |
| 13 | 1+3.29e3T+6.27e7T2 |
| 17 | 1+(2.95e3+5.10e3i)T+(−2.05e8+3.55e8i)T2 |
| 19 | 1+(3.32e3−5.75e3i)T+(−4.46e8−7.74e8i)T2 |
| 23 | 1+(991−1.71e3i)T+(−1.70e9−2.94e9i)T2 |
| 29 | 1+2.08e5T+1.72e10T2 |
| 31 | 1+(−5.88e4−1.02e5i)T+(−1.37e10+2.38e10i)T2 |
| 37 | 1+(−1.67e5+2.90e5i)T+(−4.74e10−8.22e10i)T2 |
| 41 | 1+2.65e5T+1.94e11T2 |
| 43 | 1+9.32e4T+2.71e11T2 |
| 47 | 1+(−3.28e5+5.69e5i)T+(−2.53e11−4.38e11i)T2 |
| 53 | 1+(−3.04e5−5.27e5i)T+(−5.87e11+1.01e12i)T2 |
| 59 | 1+(−2.68e5−4.64e5i)T+(−1.24e12+2.15e12i)T2 |
| 61 | 1+(−8.98e5+1.55e6i)T+(−1.57e12−2.72e12i)T2 |
| 67 | 1+(1.06e6+1.83e6i)T+(−3.03e12+5.24e12i)T2 |
| 71 | 1+1.19e6T+9.09e12T2 |
| 73 | 1+(5.28e5+9.14e5i)T+(−5.52e12+9.56e12i)T2 |
| 79 | 1+(4.99e5−8.64e5i)T+(−9.60e12−1.66e13i)T2 |
| 83 | 1−3.89e6T+2.71e13T2 |
| 89 | 1+(−2.31e6+4.00e6i)T+(−2.21e13−3.83e13i)T2 |
| 97 | 1−1.52e7T+8.07e13T2 |
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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
−9.495818987264873884522411299940, −8.785263276893203329811772806620, −7.79158005181949934800826407394, −7.02072238307116102495631711165, −5.80497468872726357241131205572, −4.96147929170139513697898724779, −3.82208079131516615931526356892, −3.02111947435136813503268888911, −1.92356497833029215926701013168, −0.41759672478793209281249335355,
0.67074930057982823709831660355, 1.85958056382846429834314100631, 2.75950054500881910774087486123, 4.02602473199963266099005443967, 4.95749975630065742813192181952, 6.06285218555691671597483061057, 7.05102207197775877800884172710, 7.81551376058665930169679895286, 8.616739772922722157718291234629, 9.526310789008828306810841051286