L(s) = 1 | + (−19.5 + 11.3i)2-s + (57.7 + 236. i)3-s + (255. − 443. i)4-s + (−2.30e3 − 1.33e3i)5-s + (−3.80e3 − 3.97e3i)6-s + (−3.51e3 − 6.08e3i)7-s + 1.15e4i·8-s + (−5.23e4 + 2.72e4i)9-s + 6.02e4·10-s + (2.13e4 − 1.23e4i)11-s + (1.19e5 + 3.48e4i)12-s + (2.70e5 − 4.68e5i)13-s + (1.37e5 + 7.95e4i)14-s + (1.81e5 − 6.21e5i)15-s + (−1.31e5 − 2.27e5i)16-s − 1.33e6i·17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.237 + 0.971i)3-s + (0.249 − 0.433i)4-s + (−0.737 − 0.425i)5-s + (−0.488 − 0.510i)6-s + (−0.209 − 0.362i)7-s + 0.353i·8-s + (−0.887 + 0.461i)9-s + 0.602·10-s + (0.132 − 0.0764i)11-s + (0.480 + 0.139i)12-s + (0.728 − 1.26i)13-s + (0.256 + 0.147i)14-s + (0.238 − 0.817i)15-s + (−0.125 − 0.216i)16-s − 0.943i·17-s + ⋯ |
Λ(s)=(=(18s/2ΓC(s)L(s)(0.608+0.793i)Λ(11−s)
Λ(s)=(=(18s/2ΓC(s+5)L(s)(0.608+0.793i)Λ(1−s)
Degree: |
2 |
Conductor: |
18
= 2⋅32
|
Sign: |
0.608+0.793i
|
Analytic conductor: |
11.4364 |
Root analytic conductor: |
3.38177 |
Motivic weight: |
10 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ18(5,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 18, ( :5), 0.608+0.793i)
|
Particular Values
L(211) |
≈ |
0.715586−0.353042i |
L(21) |
≈ |
0.715586−0.353042i |
L(6) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(19.5−11.3i)T |
| 3 | 1+(−57.7−236.i)T |
good | 5 | 1+(2.30e3+1.33e3i)T+(4.88e6+8.45e6i)T2 |
| 7 | 1+(3.51e3+6.08e3i)T+(−1.41e8+2.44e8i)T2 |
| 11 | 1+(−2.13e4+1.23e4i)T+(1.29e10−2.24e10i)T2 |
| 13 | 1+(−2.70e5+4.68e5i)T+(−6.89e10−1.19e11i)T2 |
| 17 | 1+1.33e6iT−2.01e12T2 |
| 19 | 1−3.09e6T+6.13e12T2 |
| 23 | 1+(−2.15e6−1.24e6i)T+(2.07e13+3.58e13i)T2 |
| 29 | 1+(2.67e7−1.54e7i)T+(2.10e14−3.64e14i)T2 |
| 31 | 1+(−1.67e7+2.89e7i)T+(−4.09e14−7.09e14i)T2 |
| 37 | 1+3.54e7T+4.80e15T2 |
| 41 | 1+(1.05e8+6.10e7i)T+(6.71e15+1.16e16i)T2 |
| 43 | 1+(1.18e8+2.05e8i)T+(−1.08e16+1.87e16i)T2 |
| 47 | 1+(−5.31e7+3.06e7i)T+(2.62e16−4.55e16i)T2 |
| 53 | 1−3.59e8iT−1.74e17T2 |
| 59 | 1+(8.65e8+4.99e8i)T+(2.55e17+4.42e17i)T2 |
| 61 | 1+(1.65e8+2.85e8i)T+(−3.56e17+6.17e17i)T2 |
| 67 | 1+(9.42e8−1.63e9i)T+(−9.11e17−1.57e18i)T2 |
| 71 | 1−1.92e9iT−3.25e18T2 |
| 73 | 1−2.12e9T+4.29e18T2 |
| 79 | 1+(1.67e8+2.90e8i)T+(−4.73e18+8.19e18i)T2 |
| 83 | 1+(−5.02e9+2.90e9i)T+(7.75e18−1.34e19i)T2 |
| 89 | 1+1.02e10iT−3.11e19T2 |
| 97 | 1+(−5.48e9−9.50e9i)T+(−3.68e19+6.38e19i)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
−15.99043417615301716687396368530, −15.31033934762495598283573441276, −13.68266491987183093575743699232, −11.60047655518057805341689061591, −10.26771763369224977950320858893, −8.930698387980429475022268847077, −7.64295526509202331965920368397, −5.32865880998141494207350638972, −3.47396082334519368466819144701, −0.44381490262993217299827613638,
1.54435693365863549505116885424, 3.38026199651728517675935662020, 6.51769486992374685524184672264, 7.83894131503074214544610437397, 9.173678048718372686647603869962, 11.23259114757426153247593920414, 12.13439720954650894861380422545, 13.63382701123202567601021169665, 15.18045414318985129610414619798, 16.68750241866665522610645090545