L(s) = 1 | + (0.152 − 0.263i)3-s + (17.7 + 30.7i)5-s + (121. + 210. i)9-s + (−282. + 490. i)11-s − 983.·13-s + 10.8·15-s + (100. − 173. i)17-s + (414. + 717. i)19-s + (−2.21e3 − 3.84e3i)23-s + (931. − 1.61e3i)25-s + 147.·27-s − 3.71e3·29-s + (496. − 859. i)31-s + (86.0 + 149. i)33-s + (4.17e3 + 7.23e3i)37-s + ⋯ |
L(s) = 1 | + (0.00975 − 0.0168i)3-s + (0.317 + 0.550i)5-s + (0.499 + 0.865i)9-s + (−0.704 + 1.22i)11-s − 1.61·13-s + 0.0123·15-s + (0.0839 − 0.145i)17-s + (0.263 + 0.456i)19-s + (−0.874 − 1.51i)23-s + (0.298 − 0.516i)25-s + 0.0390·27-s − 0.820·29-s + (0.0927 − 0.160i)31-s + (0.0137 + 0.0238i)33-s + (0.501 + 0.869i)37-s + ⋯ |
Λ(s)=(=(392s/2ΓC(s)L(s)(−0.701+0.712i)Λ(6−s)
Λ(s)=(=(392s/2ΓC(s+5/2)L(s)(−0.701+0.712i)Λ(1−s)
Degree: |
2 |
Conductor: |
392
= 23⋅72
|
Sign: |
−0.701+0.712i
|
Analytic conductor: |
62.8704 |
Root analytic conductor: |
7.92908 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ392(361,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 392, ( :5/2), −0.701+0.712i)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
good | 3 | 1+(−0.152+0.263i)T+(−121.5−210.i)T2 |
| 5 | 1+(−17.7−30.7i)T+(−1.56e3+2.70e3i)T2 |
| 11 | 1+(282.−490.i)T+(−8.05e4−1.39e5i)T2 |
| 13 | 1+983.T+3.71e5T2 |
| 17 | 1+(−100.+173.i)T+(−7.09e5−1.22e6i)T2 |
| 19 | 1+(−414.−717.i)T+(−1.23e6+2.14e6i)T2 |
| 23 | 1+(2.21e3+3.84e3i)T+(−3.21e6+5.57e6i)T2 |
| 29 | 1+3.71e3T+2.05e7T2 |
| 31 | 1+(−496.+859.i)T+(−1.43e7−2.47e7i)T2 |
| 37 | 1+(−4.17e3−7.23e3i)T+(−3.46e7+6.00e7i)T2 |
| 41 | 1−1.34e4T+1.15e8T2 |
| 43 | 1−298.T+1.47e8T2 |
| 47 | 1+(9.36e3+1.62e4i)T+(−1.14e8+1.98e8i)T2 |
| 53 | 1+(8.01e3−1.38e4i)T+(−2.09e8−3.62e8i)T2 |
| 59 | 1+(−6.37e3+1.10e4i)T+(−3.57e8−6.19e8i)T2 |
| 61 | 1+(1.74e4+3.02e4i)T+(−4.22e8+7.31e8i)T2 |
| 67 | 1+(5.98e3−1.03e4i)T+(−6.75e8−1.16e9i)T2 |
| 71 | 1+1.29e4T+1.80e9T2 |
| 73 | 1+(−4.05e4+7.03e4i)T+(−1.03e9−1.79e9i)T2 |
| 79 | 1+(2.34e4+4.07e4i)T+(−1.53e9+2.66e9i)T2 |
| 83 | 1−1.11e5T+3.93e9T2 |
| 89 | 1+(1.73e4+3.00e4i)T+(−2.79e9+4.83e9i)T2 |
| 97 | 1+9.26e4T+8.58e9T2 |
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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
−10.12064830872323697287756886564, −9.597116996931064262514164635335, −7.982408227232130310704255566841, −7.44109059536326093038781145247, −6.46510147920952946613431609274, −5.09779449583118466998025850919, −4.42141620430074793555735785341, −2.62133778559582340435954109874, −2.00714467699451488098206118407, 0,
1.19834042302131322409420821547, 2.67044253719599544785652232097, 3.87288310636933518219682857331, 5.15993066550952085231584634587, 5.88263983980064330325599409549, 7.20320799779790757994912182930, 7.987753568825606354849972322491, 9.323119917223335594406326369383, 9.600368967244759869663738684843