L(s) = 1 | + (1.73 + i)2-s + (0.622 − 1.07i)3-s + (1.99 + 3.46i)4-s − 20.5i·5-s + (2.15 − 1.24i)6-s + (18.4 − 10.6i)7-s + 7.99i·8-s + (12.7 + 22.0i)9-s + (20.5 − 35.5i)10-s + (−45.9 − 26.5i)11-s + 4.98·12-s + 42.5·14-s + (−22.1 − 12.7i)15-s + (−8 + 13.8i)16-s + (−34.6 − 59.9i)17-s + 50.8i·18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.119 − 0.207i)3-s + (0.249 + 0.433i)4-s − 1.83i·5-s + (0.146 − 0.0847i)6-s + (0.993 − 0.573i)7-s + 0.353i·8-s + (0.471 + 0.816i)9-s + (0.648 − 1.12i)10-s + (−1.25 − 0.727i)11-s + 0.119·12-s + 0.811·14-s + (−0.380 − 0.219i)15-s + (−0.125 + 0.216i)16-s + (−0.493 − 0.855i)17-s + 0.666i·18-s + ⋯ |
Λ(s)=(=(338s/2ΓC(s)L(s)(0.0841+0.996i)Λ(4−s)
Λ(s)=(=(338s/2ΓC(s+3/2)L(s)(0.0841+0.996i)Λ(1−s)
Degree: |
2 |
Conductor: |
338
= 2⋅132
|
Sign: |
0.0841+0.996i
|
Analytic conductor: |
19.9426 |
Root analytic conductor: |
4.46571 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ338(147,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 338, ( :3/2), 0.0841+0.996i)
|
Particular Values
L(2) |
≈ |
2.679843078 |
L(21) |
≈ |
2.679843078 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.73−i)T |
| 13 | 1 |
good | 3 | 1+(−0.622+1.07i)T+(−13.5−23.3i)T2 |
| 5 | 1+20.5iT−125T2 |
| 7 | 1+(−18.4+10.6i)T+(171.5−297.i)T2 |
| 11 | 1+(45.9+26.5i)T+(665.5+1.15e3i)T2 |
| 17 | 1+(34.6+59.9i)T+(−2.45e3+4.25e3i)T2 |
| 19 | 1+(−39.8+23.0i)T+(3.42e3−5.94e3i)T2 |
| 23 | 1+(43.6−75.5i)T+(−6.08e3−1.05e4i)T2 |
| 29 | 1+(−81.0+140.i)T+(−1.21e4−2.11e4i)T2 |
| 31 | 1+28.3iT−2.97e4T2 |
| 37 | 1+(96.9+55.9i)T+(2.53e4+4.38e4i)T2 |
| 41 | 1+(−72.9−42.1i)T+(3.44e4+5.96e4i)T2 |
| 43 | 1+(164.+284.i)T+(−3.97e4+6.88e4i)T2 |
| 47 | 1−63.2iT−1.03e5T2 |
| 53 | 1−721.T+1.48e5T2 |
| 59 | 1+(−709.+409.i)T+(1.02e5−1.77e5i)T2 |
| 61 | 1+(−198.−344.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(67.4+38.9i)T+(1.50e5+2.60e5i)T2 |
| 71 | 1+(−624.+360.i)T+(1.78e5−3.09e5i)T2 |
| 73 | 1−57.1iT−3.89e5T2 |
| 79 | 1+419.T+4.93e5T2 |
| 83 | 1−917.iT−5.71e5T2 |
| 89 | 1+(328.+189.i)T+(3.52e5+6.10e5i)T2 |
| 97 | 1+(300.−173.i)T+(4.56e5−7.90e5i)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
−11.15386940930845902546945051422, −9.958530472859566779553353352669, −8.612893949473297202781168138292, −8.036190373266643370288401685550, −7.31221906248212515468418099179, −5.44662039614328278535923451922, −5.03807172710048136000385529281, −4.13657438736487968517915652246, −2.18474614709589689505723538942, −0.75850515719202127173885847193,
1.99661022148926967086311631639, 2.93204603150945446695540407121, 4.09101468856274269720556848912, 5.33993645813843285430740969798, 6.49614208368045295120230608296, 7.30138394418491732743404718010, 8.445953302868345638747281993313, 10.03873149985349852398555174158, 10.40109520433923868808933617512, 11.30754623132320200797405910138