L(s) = 1 | + (−1.02 − 1.39i)3-s + (−0.244 − 0.423i)5-s + (−0.904 + 2.86i)9-s + (1.31 + 0.761i)11-s − 0.652i·13-s + (−0.341 + 0.775i)15-s + (3.77 − 6.54i)17-s + (0.364 − 0.210i)19-s + (−5.11 + 2.95i)23-s + (2.38 − 4.12i)25-s + (4.92 − 1.66i)27-s − 7.16i·29-s + (−6.39 − 3.69i)31-s + (−0.286 − 2.62i)33-s + (4.04 + 7.00i)37-s + ⋯ |
L(s) = 1 | + (−0.591 − 0.806i)3-s + (−0.109 − 0.189i)5-s + (−0.301 + 0.953i)9-s + (0.397 + 0.229i)11-s − 0.180i·13-s + (−0.0881 + 0.200i)15-s + (0.916 − 1.58i)17-s + (0.0836 − 0.0482i)19-s + (−1.06 + 0.616i)23-s + (0.476 − 0.824i)25-s + (0.947 − 0.320i)27-s − 1.33i·29-s + (−1.14 − 0.662i)31-s + (−0.0498 − 0.456i)33-s + (0.665 + 1.15i)37-s + ⋯ |
Λ(s)=(=(1176s/2ΓC(s)L(s)(−0.746+0.665i)Λ(2−s)
Λ(s)=(=(1176s/2ΓC(s+1/2)L(s)(−0.746+0.665i)Λ(1−s)
Degree: |
2 |
Conductor: |
1176
= 23⋅3⋅72
|
Sign: |
−0.746+0.665i
|
Analytic conductor: |
9.39040 |
Root analytic conductor: |
3.06437 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1176(521,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1176, ( :1/2), −0.746+0.665i)
|
Particular Values
L(1) |
≈ |
0.9142652921 |
L(21) |
≈ |
0.9142652921 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(1.02+1.39i)T |
| 7 | 1 |
good | 5 | 1+(0.244+0.423i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−1.31−0.761i)T+(5.5+9.52i)T2 |
| 13 | 1+0.652iT−13T2 |
| 17 | 1+(−3.77+6.54i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−0.364+0.210i)T+(9.5−16.4i)T2 |
| 23 | 1+(5.11−2.95i)T+(11.5−19.9i)T2 |
| 29 | 1+7.16iT−29T2 |
| 31 | 1+(6.39+3.69i)T+(15.5+26.8i)T2 |
| 37 | 1+(−4.04−7.00i)T+(−18.5+32.0i)T2 |
| 41 | 1+2.53T+41T2 |
| 43 | 1+5.56T+43T2 |
| 47 | 1+(4.65+8.06i)T+(−23.5+40.7i)T2 |
| 53 | 1+(1.26+0.731i)T+(26.5+45.8i)T2 |
| 59 | 1+(−3.67+6.37i)T+(−29.5−51.0i)T2 |
| 61 | 1+(9.65−5.57i)T+(30.5−52.8i)T2 |
| 67 | 1+(7.31−12.6i)T+(−33.5−58.0i)T2 |
| 71 | 1+12.8iT−71T2 |
| 73 | 1+(−8.00−4.62i)T+(36.5+63.2i)T2 |
| 79 | 1+(0.607+1.05i)T+(−39.5+68.4i)T2 |
| 83 | 1+9.03T+83T2 |
| 89 | 1+(4.77+8.27i)T+(−44.5+77.0i)T2 |
| 97 | 1−4.95iT−97T2 |
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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
−9.634726432611621213529270578886, −8.423715366853462671940624033007, −7.71643146447744692309866764238, −7.02028361401949706847650189202, −6.09809568447721236433593810332, −5.32016002597123252351415979022, −4.39854643896850132428579868633, −3.02175218455236110502640464610, −1.79047128028968487160298493196, −0.44292954919824002636692485881,
1.48997388274708261618208640141, 3.29785748944395591755135057810, 3.90551448153218601661989377158, 4.99465862450517893487424728759, 5.87060420150621987584094411771, 6.53795486884365111332551776250, 7.62824205980084766227315123141, 8.640519076375596450149843266999, 9.314120647431845621061276636893, 10.25708256349848728907225582076