L(s) = 1 | + (−0.720 − 1.24i)2-s + (2.96 − 5.13i)4-s + (2.5 + 4.33i)5-s + (18.2 + 3.22i)7-s − 20.0·8-s + (3.60 − 6.23i)10-s + (−5.12 + 8.87i)11-s + 64.1·13-s + (−9.11 − 25.0i)14-s + (−9.25 − 16.0i)16-s + (10.9 − 18.8i)17-s + (55.4 + 96.1i)19-s + 29.6·20-s + 14.7·22-s + (29.9 + 51.8i)23-s + ⋯ |
L(s) = 1 | + (−0.254 − 0.441i)2-s + (0.370 − 0.641i)4-s + (0.223 + 0.387i)5-s + (0.984 + 0.174i)7-s − 0.886·8-s + (0.113 − 0.197i)10-s + (−0.140 + 0.243i)11-s + 1.36·13-s + (−0.173 − 0.478i)14-s + (−0.144 − 0.250i)16-s + (0.155 − 0.269i)17-s + (0.670 + 1.16i)19-s + 0.331·20-s + 0.143·22-s + (0.271 + 0.470i)23-s + ⋯ |
Λ(s)=(=(315s/2ΓC(s)L(s)(0.723+0.690i)Λ(4−s)
Λ(s)=(=(315s/2ΓC(s+3/2)L(s)(0.723+0.690i)Λ(1−s)
Degree: |
2 |
Conductor: |
315
= 32⋅5⋅7
|
Sign: |
0.723+0.690i
|
Analytic conductor: |
18.5856 |
Root analytic conductor: |
4.31110 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ315(46,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 315, ( :3/2), 0.723+0.690i)
|
Particular Values
L(2) |
≈ |
2.147719713 |
L(21) |
≈ |
2.147719713 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1+(−2.5−4.33i)T |
| 7 | 1+(−18.2−3.22i)T |
good | 2 | 1+(0.720+1.24i)T+(−4+6.92i)T2 |
| 11 | 1+(5.12−8.87i)T+(−665.5−1.15e3i)T2 |
| 13 | 1−64.1T+2.19e3T2 |
| 17 | 1+(−10.9+18.8i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(−55.4−96.1i)T+(−3.42e3+5.94e3i)T2 |
| 23 | 1+(−29.9−51.8i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1−22.2T+2.43e4T2 |
| 31 | 1+(−34.9+60.4i)T+(−1.48e4−2.57e4i)T2 |
| 37 | 1+(176.+305.i)T+(−2.53e4+4.38e4i)T2 |
| 41 | 1−378.T+6.89e4T2 |
| 43 | 1−30.1T+7.95e4T2 |
| 47 | 1+(20.7+35.9i)T+(−5.19e4+8.99e4i)T2 |
| 53 | 1+(−252.+437.i)T+(−7.44e4−1.28e5i)T2 |
| 59 | 1+(−269.+466.i)T+(−1.02e5−1.77e5i)T2 |
| 61 | 1+(176.+305.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(−116.+200.i)T+(−1.50e5−2.60e5i)T2 |
| 71 | 1−723.T+3.57e5T2 |
| 73 | 1+(529.−917.i)T+(−1.94e5−3.36e5i)T2 |
| 79 | 1+(−83.4−144.i)T+(−2.46e5+4.26e5i)T2 |
| 83 | 1+924.T+5.71e5T2 |
| 89 | 1+(496.+859.i)T+(−3.52e5+6.10e5i)T2 |
| 97 | 1−744.T+9.12e5T2 |
show more | |
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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
−11.13106533737730316009423346351, −10.32400375547037583762688443785, −9.422826659486779019818717007711, −8.380582442317214275061752229335, −7.28403656645745840426146159560, −6.03712766024929907060919148688, −5.31386904898557916011420886407, −3.65565385596705709332454895149, −2.18523721576198602318542350191, −1.13648663563468330559242619539,
1.16125601582875576190518848833, 2.83082883737908951855221857088, 4.21561100997831392673047587461, 5.49451570324665608783412067722, 6.58027969511954511573682831910, 7.62401951147319552091452851653, 8.491224079194700445354153697507, 9.036417345528688412592510009269, 10.58157506024559806494580267775, 11.34403478066675768521314643329